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If a fair coin is tossed 5 times, what is the probability, to the nearest thousandth, of getting exactly 4 tails?
Answer:

If a fair coin is tossed $5$ times, what is the probability, to the nearest thousandth, of getting exactly $4$ tails? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. If a fair coin is tossed

5

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

4

tails?

\newline

Answer:

Determine probability of tail: Determine the probability of getting a tail on a single coin toss. $\newline$ A fair coin has two sides, heads and tails, so the probability of getting a tail on a single toss is $\frac{1}{2}$ .
Calculate $4$ tails in $5$ tosses: Calculate the probability of getting exactly $4$ tails in $5$ tosses. $\newline$ This is a binomial probability problem, where we want exactly $4$ successes (tails) in $5$ trials (tosses), with the probability of success on each trial being $\frac{1}{2}$ . $\newline$ The binomial probability formula is: $\newline$ $P(X = k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})$ $\newline$ where: $\newline$ - $P(X = k)$ is the probability of getting $k$ successes in $n$ trials, $\newline$ - $\binom{n}{k}$ is the number of combinations of $n$ things taken $k$ at a time, $\newline$ - $5$ $2$ is the probability of success on a single trial, $\newline$ - $5$ $3$ is the probability of failure on a single trial.
Calculate combinations of $5$ things: Calculate the number of combinations of $5$ things taken $4$ at a time. $\newline$ $(5 \choose 4) = \frac{5!}{4! \times (5-4)!} = \frac{5}{1} = 5$
Use binomial probability formula: Calculate the probability using the binomial probability formula. $\newline$ $P(X = 4) = \binom{5}{4} \cdot \left(\frac{1}{2}\right)^4 \cdot \left(\frac{1}{2}\right)^{5-4}$ $\newline$ $P(X = 4) = 5 \cdot \left(\frac{1}{2}\right)^4 \cdot \left(\frac{1}{2}\right)^1$ $\newline$ $P(X = 4) = 5 \cdot \frac{1}{16} \cdot \frac{1}{2}$ $\newline$ $P(X = 4) = \frac{5}{32}$
Convert probability to decimal: Convert the probability to a decimal to the nearest thousandth. $\newline$ $P(X = 4) = \frac{5}{32} \approx 0.15625$ $\newline$ To the nearest thousandth, this is approximately $0.156$ .

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