15 Bamey bas a biased coin. 5. Marder Probability When the coin is thrown once, It Damey throws the coin 4 limes. (a) Work out the probability that the coin will land heads exactly 3 times. \( 0.3 \times 3= \) \( 0.7 x \) (b) Work out the probability that the coin will land heads at least once. (3)

To find the probability of getting exactly 3 heads when flipping a biased coin with a probability of landing heads \( p = 0.3 \) and tails \( q = 0.7 \), you can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] Here, \( n = 4 \) (the number of flips), \( k = 3 \) (the number of heads), so: \[ P(X = 3) = \binom{4}{3} (0.3)^3 (0.7)^{1} = 4 \cdot 0.027 \cdot 0.7 = 0.0756 \] For part (b), to find the probability of getting at least one head, you can use the complement rule. First, calculate the probability of getting no heads (zero heads): \[ P(X = 0) = \binom{4}{0} (0.3)^0 (0.7)^{4} = 1 \cdot 1 \cdot 0.2401 = 0.2401 \] Now, use this to find the probability of at least one head: \[ P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2401 = 0.7599 \] So, the probability of getting exactly 3 heads is \( 0.0756 \), and the probability of getting at least one head is \( 0.7599 \).