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If a fair coin is tossed 7 times, what is the probability, to the nearest thousandth, of getting exactly 5 tails?
Answer:

If a fair coin is tossed $7$ times, what is the probability, to the nearest thousandth, of getting exactly $5$ tails? $\newline$ Answer:

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Find probabilities using the binomial distribution

Full solution

Q. If a fair coin is tossed

7

times, what is the probability, to the nearest thousandth, of getting exactly

5

tails?

\newline

Answer:

Identify Problem Type: Identify the type of probability problem. $\newline$ We are dealing with a binomial probability problem because we have a fixed number of independent trials ( $7$ coin tosses), two possible outcomes (heads or tails), and we want to find the probability of getting exactly $5$ tails.
Binomial Probability Formula: Determine the binomial probability formula. $\newline$ The binomial probability formula is $P(X = k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})$ , where: $\newline$ - $P(X = k)$ is the probability of getting $k$ successes in $n$ trials, $\newline$ - $\binom{n}{k}$ is the binomial coefficient, $\newline$ - $p$ is the probability of success on a single trial, and $\newline$ - $(1-p)$ is the probability of failure on a single trial.
Calculate Binomial Coefficient: Calculate the binomial coefficient $n \choose k$ . For our problem, $n = 7$ (number of trials) and $k = 5$ (number of successes, i.e., tails). $7 \choose 5$ = $\frac{7!}{5! \cdot (7-5)!}$ = $\frac{7!}{5! \cdot 2!}$ = $\frac{7 \cdot 6}{2 \cdot 1}$ = $\frac{42}{2}$ = $21$ .
Determine Success and Failure: Determine the probability of success $p$ and failure $1-p$ . Since the coin is fair, the probability of getting tails (success) on a single trial is $p = 0.5$ , and the probability of getting heads (failure) is also $0.5$ .
Calculate Probability of $5$ Tails: Calculate the probability of getting exactly $5$ tails. $\newline$ Using the binomial probability formula: $\newline$ $P(X = 5) = \binom{7}{5} \cdot (0.5^5) \cdot (0.5^{7-5})$ $\newline$ $P(X = 5) = 21 \cdot (0.5^5) \cdot (0.5^2)$ $\newline$ $P(X = 5) = 21 \cdot 0.03125 \cdot 0.25$ $\newline$ $P(X = 5) = 21 \cdot 0.0078125$ $\newline$ $P(X = 5) = 0.1640625$
Round Probability: Round the probability to the nearest thousandth. $P(X = 5)$ rounded to the nearest thousandth is approximately $0.164$ .

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