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(g) Practice. What is the probability of flipping at least 6 heads when you flip...

Feb 8, 2024

(g) Practice. What is the probability of flipping at least 6 heads when you flip a coin 8 times in a row? (Hint: break the event into cases)

Solution by Steps

step 1

Identify the event of interest: getting at least 6 heads in 8 coin flips

step 2

Calculate the probability of exactly 6 heads:

P(X=6) = \frac{{\text{binomial}(8, 6)}}{256}

step 3

Calculate the probability of exactly 7 heads:

P(X=7) = \frac{{\text{binomial}(8, 7)}}{256}

step 4

Calculate the probability of exactly 8 heads:

P(X=8) = \frac{{\text{binomial}(8, 8)}}{256}

step 5

Add the probabilities of getting 6, 7, and 8 heads to find the total probability of getting at least 6 heads:

P(X \geq 6) = P(X=6) + P(X=7) + P(X=8)

step 6

Use the binomial coefficient formula

\text{binomial}(n, k) = \frac{n!}{k!(n-k)!}

to calculate each probability

step 7

Compute the probabilities:

P(X=6) = \frac{28}{256}

P(X=7) = \frac{8}{256}

P(X=8) = \frac{1}{256}

step 8

Sum the probabilities:

P(X \geq 6) = \frac{28 + 8 + 1}{256}

step 9

Simplify the sum to get the final probability:

P(X \geq 6) = \frac{37}{256}

Answer

P(X \geq 6) = \frac{37}{256}

Key Concept

Binomial Probability

Explanation

The probability of flipping at least 6 heads in 8 coin flips is calculated using the binomial probability formula, considering the number of successful outcomes (heads) and the total number of flips.