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If a fair coin is tossed 7 times, what is the probability, to the nearest thousandth, of getting exactly 4 heads?
Answer:

If a fair coin is tossed $7$ times, what is the probability, to the nearest thousandth, of getting exactly $4$ heads? $\newline$ Answer:

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Full solution

Q. If a fair coin is tossed

7

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

4

heads?

\newline

Answer:

Identify Problem Type: Identify the type of probability problem. 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 $4$ heads.
Calculate Binomial Probability: Calculate the binomial probability. $\newline$ 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. $\newline$ For a fair coin, $p = 0.5$ (probability of getting heads), and $n = 7$ (number of trials).
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{7}{4}$ . $\binom{7}{4} = \frac{7!}{4! \cdot (7-4)!} = \frac{7!}{4! \cdot 3!} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35$ .
Calculate Probability: Calculate the probability of getting exactly $4$ heads. $\newline$ Using the binomial probability formula: $\newline$ $P(X = 4) = \binom{7}{4} \times (0.5)^4 \times (0.5)^{7-4}$ $\newline$ $P(X = 4) = 35 \times (0.5)^4 \times (0.5)^3$ $\newline$ $P(X = 4) = 35 \times (0.5)^7$ $\newline$ $P(X = 4) = 35 \times (\frac{1}{128})$ $\newline$ $P(X = 4) = \frac{35}{128}$ $\newline$ $P(X = 4) = 0.2734375$
Round to Nearest Thousandth: Round the probability to the nearest thousandth. $P(X = 4)$ rounded to the nearest thousandth is approximately $0.273$ .

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