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Math  /  Data & Statistics

QuestionIf a fair coin is tossed 4 times, what is the probability, to the nearest thousandth, of getting exactly 1 heads?

Studdy Solution

STEP 1

1. The coin is fair, meaning the probability of heads (H) is 0.50.5 and the probability of tails (T) is 0.50.5.
2. Each coin toss is independent.
3. We are using the binomial probability formula to find the probability of exactly 1 heads in 4 tosses.

STEP 2

1. Identify the parameters for the binomial distribution.
2. Apply the binomial probability formula.
3. Calculate the probability and round to the nearest thousandth.

STEP 3

Identify the parameters for the binomial distribution: - Number of trials n=4 n = 4 - Probability of success (getting heads) p=0.5 p = 0.5 - Number of successes k=1 k = 1

STEP 4

Apply the binomial probability formula:
The probability of getting exactly k k successes (heads) in n n trials is given by:
P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
For this problem:
P(X=1)=(41)(0.5)1(1−0.5)4−1P(X = 1) = \binom{4}{1} (0.5)^1 (1-0.5)^{4-1}

STEP 5

Calculate the binomial coefficient:
(41)=4!1!(4−1)!=4\binom{4}{1} = \frac{4!}{1!(4-1)!} = 4

STEP 6

Calculate the probability:
P(X=1)=4×(0.5)1×(0.5)3=4×0.5×0.125=0.25P(X = 1) = 4 \times (0.5)^1 \times (0.5)^3 = 4 \times 0.5 \times 0.125 = 0.25

STEP 7

Round the probability to the nearest thousandth:
0.25=0.2500.25 = 0.250
The probability of getting exactly 1 heads in 4 tosses is:
0.250 \boxed{0.250}

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