Question

a coin flip determines who gets the ball first at the beginning of a football game, with the visiting team calling heads or tails. the captain of one particular team always calls heads. in the first four games as visitor of a season, find the probability that his team
(a) wins the toss one time.
(b) loses the toss four times.
(c) wins the toss more than three times.
(d) loses the toss no more than three times.
(e) loses the toss at least once.
write your answers in exact, simplified form.
part 1 of 5
(a) the probability that the team wins the toss one time is

part 2 of 5
(b) the probability that the team loses the toss all four times is

Explanation:

Step1: Calculate the probability of winning a single toss

The probability of winning a single coin - toss (calling it correctly) is \(p=\frac{1}{2}\), and the probability of losing is \(q = 1 - p=\frac{1}{2}\). We use the binomial probability formula \(P(X = k)=C(n,k)\times p^{k}\times q^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\) and \(n = 4\) (number of coin - tosses).

Step2: Solve for part (a) (wins the toss one time)

For \(n = 4\), \(k = 1\):

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Step3: Solve for part (b) (loses the toss all four times)

If losing the toss all four times, it means winning \(k = 0\) times.

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Answer:

(a) \(\frac{1}{4}\)
(b) \(\frac{1}{16}\)