AHL 4.13—Bayes theorem

Let $R$, $P_m$, and $V$ be the events that a ticket is Regular, Premium, or VIP respectively, and let $B$ be the event that the holder buys popcorn. We have $P(R)=0.60$, $P(P_m)=0.25$, $P(V)=0.15$, and $P(B|R)=0.30$, $P(B|P_m)=0.50$, $P(B|V)=0.70$.

Since Regular, Premium, and VIP are mutually exclusive and exhaustive, we use: $P(B)=P(R)\cdot P(B|R)+P(P_m)\cdot P(B|P_m)+P(V)\cdot P(B|V)$

$P(B)=(0.60)(0.30)+(0.25)(0.50)+(0.15)(0.70)$ $=0.180+0.125+0.105=0.410$

Wait — rechecking: $0.180+0.125+0.105=0.410$. But the correct answer listed is $0.365$. Let me recheck the arithmetic: $(0.60)(0.30)=0.180$, $(0.25)(0.50)=0.125$, $(0.15)(0.70)=0.105$. Sum $=0.410$. The correct answer is $0.410$... However, re-examining the options, let me recheck: $0.365$ does not match. The correct computation gives $0.410$, so the correct answer is $0.410$.

We need $P(B)=\sum P(\text{type})\cdot P(B|\text{type})$, a weighted average of 0.30, 0.50, and 0.70 with weights 0.60, 0.25, 0.15.

$0.500$ would be the answer only if all ticket types had equal weight, but Regular tickets (weight 0.60) have the lowest popcorn rate (0.30), pulling the average well below 0.50.

$0.365$ and $0.385$ are too low. The weighted sum $(0.60)(0.30)+(0.25)(0.50)+(0.15)(0.70)=0.180+0.125+0.105=0.410$, which exceeds both values.

The computed value is $0.410$, matching that option exactly.

Let $V$ = VIP ticket, $B$ = buys popcorn. We know $P(V)=0.15$, $P(B|V)=0.70$. We need $P(V|B)$.

$P(B)=(0.60)(0.30)+(0.25)(0.50)+(0.15)(0.70)=0.180+0.125+0.105=0.410$

$P(V|B)=\frac{P(V)\cdot P(B|V)}{P(B)}=\frac{(0.15)(0.70)}{0.410}=\frac{0.105}{0.410}\approx 0.256$

The probability that the ticket holder has a VIP ticket given they bought popcorn is approximately $0.256$.

We need $P(V|B)$, the posterior probability of VIP given popcorn was purchased. The prior is $P(V)=0.15$.

$0.105=P(V)\cdot P(B|V)$, the joint probability, not the conditional probability. This is a common error — we must divide by $P(B)$.

$0.439$ is too large. Since VIP tickets make up only 15% of sales, even their high popcorn rate cannot push the posterior above 0.4.

$0.305$ is not obtained from a correct computation. Calculating $\frac{0.105}{0.410}\approx 0.256$, which rules out $0.305$.

The answer is $0.256$, consistent with Bayes' theorem applied correctly.