Step 1 :The coin flip is a binomial experiment where there are only two outcomes - heads or tails. The probability of getting heads (winning the toss) or tails (losing the toss) is 0.5 each.
Step 2 :For part (a), we need to find the probability of winning the toss exactly 3 times in 4 trials. This is a binomial probability problem which can be solved using the formula: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes in n trials, \(C(n, k)\) is the combination of n items taken k at a time, p is the probability of success on a single trial, n is the number of trials, k is the number of successes.
Step 3 :Substituting the given values into the formula, we get \(P(X=3) = C(4, 3) * (0.5^3) * ((1-0.5)^(4-3))\). Simplifying this, we get a probability of 0.25.
Step 4 :For part (b), we need to find the probability of losing the toss all four times. This is simply the probability of getting tails 4 times in a row, which can be calculated as \((0.5)^4\).
Step 5 :Simplifying this, we get a probability of 0.0625.
Step 6 :Final Answer: (a) The probability that the team wins the toss three times is \(\boxed{0.25}\). (b) The probability that the team loses the toss all four times is \(\boxed{0.0625}\).