15 Barney has a biased coin. When the coin is thrown once, the probability that tho coin will land heads is 0.3 Bamey throws the coin 4 times. (a) Work out the probability that the coin will land heads exactly 3 times. (b) Work out the probability that the coin will lard heads at least once. (b) Coblity

To solve the problem, we will use the binomial probability formula, which is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( n \) is the number of trials (coin tosses), - \( k \) is the number of successful outcomes (heads), - \( p \) is the probability of success on a single trial, - \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \). ### Given: - Probability of heads, \( p = 0.3 \) - Probability of tails, \( 1 - p = 0.7 \) - Number of trials, \( n = 4 \) ### (a) Probability of getting exactly 3 heads We need to find \( P(X = 3) \): \[ P(X = 3) = \binom{4}{3} (0.3)^3 (0.7)^{4-3} \] Calculating the binomial coefficient: \[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4 \times 3!}{3! \times 1!} = 4 \] Now substituting into the formula: \[ P(X = 3) = 4 \times (0.3)^3 \times (0.7)^1 \] Calculating \( (0.3)^3 \) and \( (0.7)^1 \): \[ (0.3)^3 = 0.027 \] \[ (0.7)^1 = 0.7 \] Now substituting these values back: \[ P(X = 3) = 4 \times 0.027 \times 0.7 \] Calculating this: \[ P(X = 3) = 4 \times 0.0189 = 0.0756 \] ### (b) Probability of getting at least 1 head To find the probability of getting at least one head, we can use the complement rule: \[ P(\text{at least 1 head}) = 1 - P(\text{no heads}) \] First, we calculate \( P(X = 0) \): \[ P(X = 0) = \binom{4}{0} (0.3)^0 (0.7)^{4} \] Calculating the binomial coefficient: \[ \binom{4}{0} = 1 \] Now substituting into the formula: \[ P(X = 0) = 1 \times (0.3)^0 \times (0.7)^4 \] Calculating \( (0.7)^4 \): \[ (0.7)^4 = 0.2401 \] Thus, \[ P(X = 0) = 0.2401 \] Now, using the complement rule: \[ P(\text{at least 1 head}) = 1 - P(X = 0) = 1 - 0.2401 = 0.7599 \] ### Final Answers: (a) The probability that the coin will land heads exactly 3 times is \( 0.0756 \). (b) The probability that the coin will land heads at least once is \( 0.7599 \).