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If a fair coin is tossed 4 times, what is the probability, rounded to the nearest thousandth, of getting at least 3 tails?
Answer:

If a fair coin is tossed $4$ times, what is the probability, rounded to the nearest thousandth, of getting at least $3$ tails? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. If a fair coin is tossed

4

times, what is the probability, rounded to the nearest thousandth, of getting at least

3

tails?

\newline

Answer:

Calculate Probabilities: To solve this problem, we need to calculate the probability of getting exactly $3$ tails and the probability of getting exactly $4$ tails, then add these probabilities together.
Binomial Probability Formula: The probability of getting exactly $3$ tails can be calculated using the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Probability of $3$ Tails: For exactly $3$ tails, $n=4$ , $k=3$ , and $p=0.5$ (since the coin is fair). The binomial coefficient $(4 \choose 3)$ is $4$ , because there are $4$ different ways to get $3$ tails in $4$ tosses (TTTH, THTT, HTTT, TTTT).
Calculate $3$ Tails Probability: Now we calculate the probability of getting exactly $3$ tails: $P(3 \text{ tails}) = \binom{4}{3} \times (0.5)^3 \times (0.5)^{4-3} = 4 \times (0.5)^3 \times (0.5)^1 = 4 \times 0.125 \times 0.5 = 0.25$ .
Probability of $4$ Tails: Next, we calculate the probability of getting exactly $4$ tails. For this, $n=4$ , $k=4$ , and $p=0.5$ . The binomial coefficient $(4 \choose 4)$ is $1$ , because there is only $1$ way to get $4$ tails in $4$ tosses (TTTT).
Calculate $4$ Tails Probability: Now we calculate the probability of getting exactly $4$ tails: $P(4 \text{ tails}) = \binom{4}{4} \times (0.5)^4 \times (0.5)^{4-4} = 1 \times (0.5)^4 \times (0.5)^0 = 1 \times 0.0625 \times 1 = 0.0625$ .
Add Probabilities: To find the probability of getting at least $3$ tails, we add the probabilities of getting exactly $3$ tails and exactly $4$ tails: $P(\text{at least 3 tails}) = P(3 \text{ tails}) + P(4 \text{ tails}) = 0.25 + 0.0625 = 0.3125$ .
Final Probability Calculation: Finally, we round the probability to the nearest thousandth: $P(\text{at least 3 tails}) \approx 0.313$ .

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