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7 problems match “conditional probability”. Clear

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1. Conditional probabilities from given data ●○○○○ 10 ptsGiven $P(E)=0.5$, $P(F)=0.7$ and $P(E \cap F)=0.3$, find $P(E\mid F)$ and $P(F\mid E)$. Round each answer to two decimal places.
2. Union probability from given values ●○○○○ 10 ptsGiven $P(A)=0.6$, $P(B)=0.5$ and $P(A\mid B)=0.3$, find $P(A\cup B)$. Give your answer as a decimal.
3. Ace then king without replacement ●○○○○ 10 ptsA standard 52-card deck is shuffled. One card is drawn and not replaced, then a second card is drawn. Find the probability that the first card is an ace and the second card is a king. Give your answer as a decimal rounded to four decimal places.
4. Aces before the joker ●●○○○ 20 ptsA standard 52-card deck plus one joker (53 cards in total) is shuffled into a uniformly random order and turned over one card at a time. What is the probability that all 4 aces are turned over before the joker? Give your answer as a decimal.
5. Combining four experts about a label ●●○○○ 20 ptsFour independent experts have probabilities $p_1=0.8$, $p_2=0.7$, $p_3=0.6$ and $p_4=0.5$ of being correct about a binary label $L\in\{+1,-1\}$, with a uniform prior on $L$. Expert 1 reports $L_1=-1$, expert 2 reports $L_2=+1$ and expert 3 reports $L_3=+1$ (expert 4's report is not given). Using the classifier $C(S)=\sum_i \frac{L_i}{2}\log\left(\frac{p_i}{1-p_i}\right)$, find $\Pr(L=+1\mid S)$. Round to two decimal places.
6. Last passenger in assigned seat ●●●○○ 50 ptsOne hundred passengers board a full flight in order, each with an assigned seat. The first passenger loses their boarding pass and picks a seat at random. Each later passenger sits in their assigned seat if it is free, and otherwise picks a seat at random from the remaining free seats. What is the probability that the last passenger sits in their assigned seat? Give the answer as a decimal.
7. Red balls after discarding draws ●●●○○ 50 ptsAn urn contains 2 yellow balls, 3 blue balls and 22 red balls. One ball is drawn, found to be red, and discarded. Then 5 balls are drawn and thrown away without looking at their colours. Given this information, what is the probability that if you now draw four balls from the urn, they are all red? Round to three decimal places.