Probability › Events and basic probability

Events, unions and complements

Solve the first problem 17 problems

These are 8 of the 17 problems. Sign in for free to see them all. Sign in

1. Union of two events on a loaded die ●○○○○ 10 ptsA loaded (unfair) die is tossed. Let $A$ be the event "an even number is observed" and $B$ be the event "a number less than 3 is observed". Given $P(A)=0.4$, $P(B)=0.3$ and $P(A\cap B)=0.1$, find $P(A\cup B)$.
2. 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.
3. 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.
4. Probability of A from two unions ●●○○○ 20 ptsGiven $P(A \cup B)=0.7$ and $P(A \cup B')=0.9$, determine $P(A)$.
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. False propositions about events ●●○○○ 20 ptsConsider the following five propositions. ① Toss a coin twice. Let event $A$ be "both tosses land heads" and event $B$ be "both tosses land tails"; then $A$ and $B$ are complementary events. ② In proposition ①, $A$ and $B$ are mutually exclusive events. ③ Among 10 products, 3 are defective. Three products are selected at random. Let event $A$ be "at most 2 of the selected products are defective" and event $B$ be "at least 2 of the selected products are defective"; then $A$ and $B$ are mutually exclusive events. ④ If events $A$ and $B$ satisfy $P(A)+P(B)=1$, then $A$ and $B$ are complementary events. ⑤ If $A$ and $B$ are mutually exclusive events, then $\overline{A}\cup\overline{B}$ is a certain event. Identify all the false propositions among these five and give their numbers, in increasing order.
7. Rebid probability for close bids ●●●○○ 50 ptsTwo construction companies make bids $X$ and $Y$ on a remodeling project. The joint density of $X$ and $Y$ is uniform on the rectangle $2<x<2.5$, $2<y<2.3$. If $X$ and $Y$ are within 0.1 of each other, the companies are asked to rebid; otherwise the lower bidder gets the contract. Find the probability that the companies are asked to rebid. Round to two decimal places.
8. Smallest p for triple intersection ●●●○○ 50 ptsLet events $A$, $B$ and $C$ satisfy $P(A)=P(B)=P(C)=p$. Find the infimum of the values of $p$ for which $P(A\cap B\cap C)>0.95$ holds for every possible joint distribution of $A$, $B$ and $C$. Round to four decimal places.

More in Events and basic probability

Experimental probabilitySpinners