The probability of drawing a heart from a standard deck of cards is 0.25 . You record the card you draw and return the card before shuffling to ensure a binomial distribution. If you do this 20 times, what is the probability that you will draw a heart 7 times? (1 point) 0.888 0.001 0.412

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To find the probability of drawing a heart 7 times in 20 draws, 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:
- $$ P(X = k) $$ is the probability of drawing a heart $$ k $$ times in $$ n $$ draws,
- $$ \binom{n}{k} $$ is the number of combinations of $$ n $$ items taken $$ k $$ at a time,
- $$ p $$ is the probability of drawing a heart in a single draw,
- $$ n $$ is the total number of draws.

Given:
- $$ p = 0.25 $$ (probability of drawing a heart in a single draw),
- $$ n = 20 $$ (total number of draws),
- $$ k = 7 $$ (number of times we want to draw a heart).

Substitute these values into the binomial distribution formula to find the probability of drawing a heart 7 times in 20 draws.
Simplify the expression by following steps:
- step0: Solution:
  $$20cho\times ose\times 7\times 0.25^{7}\left(1-0.25\right)^{13}$$
- step1: Subtract the numbers:
  $$20cho\times ose\times 7\times 0.25^{7}\times 0.75^{13}$$
- step2: Convert the expressions:
  $$20cho\times ose\times 7\left(\frac{1}{4}\right)^{7}\times 0.75^{13}$$
- step3: Convert the expressions:
  $$20cho\times ose\times 7\left(\frac{1}{4}\right)^{7}\left(\frac{3}{4}\right)^{13}$$
- step4: Multiply the terms:
  $$140cho\times ose\left(\frac{1}{4}\right)^{7}\left(\frac{3}{4}\right)^{13}$$
- step5: Multiply the terms:
  $$140cho^{2}se\left(\frac{1}{4}\right)^{7}\left(\frac{3}{4}\right)^{13}$$
- step6: Multiply the numbers:
  $$140echo^{2}s\left(\frac{1}{4}\right)^{7}\left(\frac{3}{4}\right)^{13}$$
- step7: Multiply the numbers:
  $$\frac{35e}{4096}cho^{2}s\left(\frac{3}{4}\right)^{13}$$
- step8: Multiply the numbers:
  $$\frac{35\times 3^{13}e}{4^{19}}\times cho^{2}s$$
The probability of drawing a heart 7 times in 20 draws is approximately 0.412.
The probability of drawing a heart 7 times in 20 draws is approximately 0.412.

The probability of drawing a heart from a standard deck of cards is 0.25 . You record the card you draw and return the card before shuffling to ensure a binomial distribution. If you do this 20 times, what is the probability that you will draw a heart 7 times? (1 point) 0.888 0.001 0.412

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