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56 problems match “probability”. Clear

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.
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.
Same heads from one and two coins ●●○○○ 20 ptsKeiko tosses one penny and Ephraim tosses two pennies. All pennies are fair. What is the probability that Ephraim gets the same number of heads that Keiko gets? Give your answer as a decimal.
Percentage of pitch times in an interval ●●○○○ 20 ptsTommy Wait's times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of 2.5 seconds. What percentage of his times between pitches are between 30 and 34 seconds? Give the answer as a percentage, rounded to two decimal places.
Largest component reliability under budget ●●○○○ 20 ptsA component's cost is $C = (10P)^2$ dollars, where $P$ is the probability that the component operates as expected. A system has two identical components, and both must operate for the system to operate. A total of $173 can be spent on the two components. What is the largest component probability $P$ that can be achieved? Round to two decimal places.
Smallest group with a shared birthday ●●●○○ 50 ptsWhat is the smallest number of people in a group such that the probability that at least two of them share the same birthday is greater than 50%? Assume there are 365 equally likely birthdays and ignore leap years.
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.
Expected count of numbers exceeding a neighbor ●●●○○ 50 ptsThe numbers $1$ to $10$ are written in a circle in a random order. Find the expected number of numbers that are at least $2$ larger than at least one of their two adjacent numbers. Round to two decimal places.
Rolling more 1s than 6s ●●●○○ 50 ptsIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1s than 6s? Round to two decimal places.
Each player gets one ace ●●●○○ 50 ptsA deck of 52 cards containing four aces is shuffled thoroughly and the cards are dealt among four players so that each player receives 13 cards. Find the probability that each player receives exactly one ace. Give the answer as a percentage rounded to one decimal place.