(b) (5%) If the tulip is red, find the conditional probability that a bag having 15 red and 10 yellow tulips was selected.

(Some critical values $F_\alpha(df_1, df_2)$ and $t_\alpha(df)$, where $\Pr(F > F_\alpha(df_1, df_2)) = \alpha$ and $\Pr(t > t_\alpha(df)) = \alpha$, are given as follows: $$ \begin{array}{l} F_{0.02}(24, 29) = 2.236, F_{0.01}(24, 29) = 2.495, F_{0.02}(29, 24) = 2.306, \\ F_{0.01}(29, 24) = 2.589, F_{0.02}(30, 25) = 2.267, F_{0.01}(30, 25) = 2.538, \\ t_{0.01}(53) = 2.398, t_{0.01}(54) = 2.397, t_{0.01}(55) = 2.396, \\ t_{0.005}(53) = 2.672, t_{0.005}(54) = 2.670, t_{0.005}(55) = 2.668 \end{array} $$ 3. Let $X_1, X_2, \ldots$, and $X_n$ be a random sample from distributions with the following probability density functions. In each case find the maximum likelihood estimator $\hat{\theta}$. (a) $(5\%) f(x; \theta) = \theta x^{\theta - 1}, 0 < x < 1, 0 < \theta < \infty$

(b) \((5\%)\) If the tulip is red, find the conditional probability that a bag having 15 red and 10 yellow tulips was selected.

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