You flip a coin 10 times. Knowing that the event satisfies the requirements for a binomial distribution, find the probability that exactly 7 of the outcomes are heads. (1 point) 0.120 0.001 0.117

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UpStudy AI · Accepted Answer

To find the probability of getting exactly 7 heads in 10 coin flips, we can use the binomial distribution formula.

The binomial distribution formula is given by:
$$ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} $$

Where:
- $$ n $$ is the number of trials (in this case, 10 coin flips),
- $$ k $$ is the number of successful outcomes (in this case, 7 heads),
- $$ p $$ is the probability of success on a single trial (in this case, 0.5 for a fair coin).

Substitute the values into the formula:
$$ P(X = 7) = \binom{10}{7} \times 0.5^7 \times (1-0.5)^{10-7} $$

Now, we can calculate the probability using the binomial distribution formula.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{10}C_{7}\times 0.5^{7}\left(1-0.5\right)^{10-7}$$
- step1: Subtract the numbers:
  $$ { }_{10}C_{7}\times 0.5^{7}\times 0.5^{10-7}$$
- step2: Subtract the numbers:
  $$ { }_{10}C_{7}\times 0.5^{7}\times 0.5^{3}$$
- step3: Expand the expression:
  $$\frac{10!}{7!\times \left(10-7\right)!}\times 0.5^{7}\times 0.5^{3}$$
- step4: Subtract the numbers:
  $$\frac{10!}{7!\times 3!}\times 0.5^{7}\times 0.5^{3}$$
- step5: Reduce the fraction:
  $$\frac{10\times 9\times 8}{3!}\times 0.5^{7}\times 0.5^{3}$$
- step6: Multiply:
  $$\frac{720}{3!}\times 0.5^{7}\times 0.5^{3}$$
- step7: Divide the terms:
  $$120\times 0.5^{7}\times 0.5^{3}$$
- step8: Convert the expressions:
  $$120\left(\frac{1}{2}\right)^{7}\times 0.5^{3}$$
- step9: Convert the expressions:
  $$120\left(\frac{1}{2}\right)^{7}\left(\frac{1}{2}\right)^{3}$$
- step10: Multiply the terms:
  $$120\left(\frac{1}{2}\right)^{7+3}$$
- step11: Add the numbers:
  $$120\left(\frac{1}{2}\right)^{10}$$
- step12: Evaluate the power:
  $$120\times \frac{1}{1024}$$
- step13: Multiply:
  $$\frac{15}{128}$$
The probability of getting exactly 7 heads in 10 coin flips is $$ \frac{15}{128} $$ or approximately 0.1171875.

Therefore, the correct answer is 0.117.
The probability of getting exactly 7 heads in 10 coin flips is approximately 0.117.

You flip a coin 10 times. Knowing that the event satisfies the requirements for a binomial distribution, find the probability that exactly 7 of the outcomes are heads. (1 point) 0.120 0.001 0.117

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