$$ P( $$ no 2 people out of 5 share same What is the probability that in a class of 60 students, at least one pair of people share the same birthday?

Question

UpStudy AI · Accepted Answer

To solve the problem of finding the probability that at least one pair of people in a class of 60 students shares the same birthday, we can use the complement principle.

1. **Known Conditions:**
   - Total number of students, $$ n = 60 $$
   - Total number of possible birthdays (assuming 365 days in a year), $$ d = 365 $$

2. **Complement Principle:**
   - First, we calculate the probability that no two students share the same birthday.
   - Then, we subtract this probability from 1 to find the probability that at least one pair shares a birthday.

3. **Calculating the Probability that No Two Students Share a Birthday:**
   - The first student can have any of the 365 days as their birthday.
   - The second student can have any of the remaining 364 days.
   - The third student can have any of the remaining 363 days, and so on.

The probability that all 60 students have different birthdays can be calculated as follows:

$$
   P(\text{no shared birthdays}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \ldots \times \frac{365 - 59}{365}
   $$

This can be expressed as:

$$
   P(\text{no shared birthdays}) = \frac{365!}{(365 - 60)! \cdot 365^{60}}
   $$

4. **Calculating the Probability of At Least One Shared Birthday:**
   - Finally, the probability that at least one pair of students shares a birthday is:

$$
   P(\text{at least one shared birthday}) = 1 - P(\text{no shared birthdays})
   $$

Now, let's calculate $$ P(\text{no shared birthdays}) $$ and then find $$ P(\text{at least one shared birthday}) $$.

I'll perform the calculation now.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(365!\right)}{\left(\left(365-60\right)!\times 365^{60}\right)}$$
- step1: Remove the parentheses:
  $$\frac{365!}{\left(365-60\right)!\times 365^{60}}$$
- step2: Subtract the numbers:
  $$\frac{365!}{305!\times 365^{60}}$$
- step3: Multiply the terms:
  $$\frac{365!}{365^{60}\times 305!}$$
- step4: Factor the expression:
  $$\frac{365\times 364!}{365^{60}\times 305!}$$
- step5: Reduce the fraction:
  $$\frac{364!}{365^{59}\times 305!}$$
- step6: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 360\times 359\times 358\times 357\times 356\times 355\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306\times 305!}{365^{59}\times 305!}$$
- step7: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 360\times 359\times 358\times 357\times 356\times 355\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{365^{59}}$$
- step8: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 5\times 72\times 359\times 358\times 357\times 356\times 355\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{365^{59}}$$
- step9: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 5\times 72\times 359\times 358\times 357\times 356\times 355\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{59}\times 73^{59}}$$
- step10: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 355\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{58}\times 73^{59}}$$
- step11: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 5\times 71\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{58}\times 73^{59}}$$
- step12: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 350\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{57}\times 73^{59}}$$
- step13: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 25\times 14\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{57}\times 73^{59}}$$
- step14: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 345\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{55}\times 73^{59}}$$
- step15: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 5\times 69\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{55}\times 73^{59}}$$
- step16: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 340\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{54}\times 73^{59}}$$
- step17: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 5\times 68\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{54}\times 73^{59}}$$
- step18: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 335\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{53}\times 73^{59}}$$
- step19: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 5\times 67\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{53}\times 73^{59}}$$
- step20: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 330\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{52}\times 73^{59}}$$
- step21: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 5\times 66\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{52}\times 73^{59}}$$
- step22: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 325\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{51}\times 73^{59}}$$
- step23: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 25\times 13\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{51}\times 73^{59}}$$
- step24: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 320\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{49}\times 73^{59}}$$
- step25: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 5\times 64\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{49}\times 73^{59}}$$
- step26: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 64\times 319\times 318\times 317\times 316\times 315\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{48}\times 73^{59}}$$
- step27: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 64\times 319\times 318\times 317\times 316\times 5\times 63\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{48}\times 73^{59}}$$
- step28: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 64\times 319\times 318\times 317\times 316\times 63\times 314\times 313\times 312\times 311\times 310\times 309\times 308\times 307\times 306}{5^{47}\times 73^{59}}$$
- step29: Factor the expression:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 64\times 319\times 318\times 317\times 316\times 63\times 314\times 313\times 312\times 311\times 5\times 62\times 309\times 308\times 307\times 306}{5^{47}\times 73^{59}}$$
- step30: Reduce the fraction:
  $$\frac{364\times 363\times 362\times 361\times 72\times 359\times 358\times 357\times 356\times 71\times 354\times 353\times 352\times 351\times 14\times 349\times 348\times 347\times 346\times 69\times 344\times 343\times 342\times 341\times 68\times 339\times 338\times 337\times 336\times 67\times 334\times 333\times 332\times 331\times 66\times 329\times 328\times 327\times 326\times 13\times 324\times 323\times 322\times 321\times 64\times 319\times 318\times 317\times 316\times 63\times 314\times 313\times 312\times 311\times 62\times 309\times 308\times 307\times 306}{5^{46}\times 73^{59}}$$
- step31: Simplify:
  $$\frac{2236978388849428224\times 9194926154160434688\times 8072737123088123136\times 1317905337264804288\times 431294849762285376\times 4535626073351611392\times 303782277313816704\times 554320177488}{5^{46}\times 73^{59}}$$
Calculate or simplify the expression $$ 1 - 0.005877 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1-0.005877$$
- step1: Subtract the numbers:
  $$0.994123$$
1. **Calculating the Probability of No Shared Birthdays:**
   - The probability that no two students share the same birthday in a class of 60 students is approximately $$ P(\text{no shared birthdays}) \approx 0.005877 $$.

2. **Calculating the Probability of At Least One Shared Birthday:**
   - Using the complement principle:
   $$
   P(\text{at least one shared birthday}) = 1 - P(\text{no shared birthdays}) \approx 1 - 0.005877 \approx 0.994123
   $$

Thus, the probability that in a class of 60 students, at least one pair of people shares the same birthday is approximately **0.9941** or **99.41%**.
The probability that at least one pair of people in a class of 60 students shares the same birthday is approximately 99.41%.

\( P( \) no 2 people out of 5 share same What is the probability that in a class of 60 students, at least one pair of people share the same birthday?

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