QuestionGiven that $X \sim B(25,0.65)$, find each of the following probabilities. Give results accurate to at least 4 decimal places. $P(X=19)$ $\square$ $P(X \leq 14)$ $\square$ $P(X \geq 12)$ $\square$

Studdy Solution
Use the cumulative distribution function to find $P(X \geq 12)$.
$$P(X \geq 12) = 1 - P(X \leq 11)$$
Calculate the cumulative probability for $P(X \leq 11)$:
$$P(X \leq 11) = \sum_{k=0}^{11} \binom{25}{k} (0.65)^k (0.35)^{25-k}$$
Then calculate:
$$P(X \geq 12) = 1 - P(X \leq 11)$$
The probabilities are:
$$P(X = 19) \approx 0.0907$$
$$P(X \leq 14) \approx 0.2123$$
$$P(X \geq 12) \approx 0.8960$$