In the following problems, the probability distributions for the parameter \(p\) of a Bernoulli process are given. Find the probability of event \(H\).

(a) \(p \sim \beta_{(12,17)}(p) . H=\mathrm{T}\).

(b) \(p \sim \beta_{(12,17)}(p) . H=\perp\).

(c) \(p \sim \beta_{(12,17)}(p) . H=\perp \mathrm{T}\).

(d) \(p \sim \beta_{(12,17)}(p) . H=\mathrm{T} \perp\).

(e) \(p \sim \beta_{(12,17)}(p) . H=\mathrm{TTT} \perp \perp \mathrm{T} \perp \perp\).

(f) \(p \sim \beta_{(12,17)}(p) . H=\perp \perp \perp \perp\) TTTT.

(g) \(p \sim \beta_{(12,17)}(p) . H=4 \perp\) and \(4 \mathrm{~T}\) (combination).

(h) \(p \sim \beta_{(52,12)}(p) . H=\) a given sequence with \(3 \top\) and \(1 \perp\).

(i) \(p \sim \beta_{(52,12)}(p)\). \(H=\) the combination \(3 \mathrm{~T}\) and \(1 \perp\).

(j) \(p \sim \beta_{(52,12)}(p)\). \(H=\) the combination \(4 \mathrm{~T}\) and \(0 \perp\).

(k) \(p \sim \beta_{(52,12)}(p)\). \(H=\) the combination \(2 \mathrm{~T}\) and \(2 \perp\).

(I) \(p \sim \beta_{(52,12)}(p)\). \(H=\) the combination \(1 \mathrm{~T}\) and \(3 \perp\)

(m) \(p \sim \beta_{(52,12)}(p) . H=\) the combination \(0 \top\) and \(4 \perp\).

(n) \(p \sim \beta_{(52,12)}(p)\). \(H=\) at least \(3 \mathrm{~T}\) in 4 trials.

(o) \(p \sim \beta_{(52,12)}(p)\). \(H=\) less than \(3 \mathrm{~T}\) in 4 trials.