4.11 Conditional probability and independence questions for IB Maths AA

Find $\mathrm{P}(A\cap B)$.

Find $\mathrm{P}(B)$.

Find $\mathrm{P}(B\mid A^{\mathrm{\prime }})$.

Given that $\mathrm{P}(G\mid S)=0.36$, find the value of $x$ and the value of $y$.

Find the probability that the visitor holds a student ticket, given that the visitor hired an audio guide.

Determine whether the events $G$ and $S$ are independent. Justify your answer.

Write down $\mathrm{P}(H_3\mid D_1)$, $\mathrm{P}(H_3\mid D_2)$ and $\mathrm{P}(H_3\mid D_3)$.

Show that $\mathrm{P}(H_3)={\displaystyle \frac{1+q}{3}}$.

Show that $\mathrm{P}(D_1\mid H_3)={\displaystyle \frac{q}{1+q}}$.

Find $\mathrm{P}(D_2\mid H_3)$ in terms of $q$.

Consider the case $q=\frac{1}{2}$. The host opens door 3. Find the probability that the contestant wins the prize if they switch to door 2.

Show that, when the host opens door 3, switching to door 2 is never less likely to win the prize than staying with door 1. State the value of $q$ for which the two choices are equally likely to win.

The contestant decides, before the game starts, always to switch. By finding $\mathrm{P}(D_3\mid H_2)$, where $H_2$ is the event that the host opens door 2, show that the probability that the contestant wins the prize is $\frac{2}{3}$ for every value of $q$.

Find the probability that the game is abandoned.

This host opens door 3 and it does not hide the prize. Show that the probability that the prize is behind door 1 is now $\frac{1}{2}$.

By comparing your answers to parts (d) and (h), explain briefly why the host's knowledge matters to the contestant.

Determine whether the events $C$ and $O$ are independent. Justify your answer.

Find the probability that the student belongs to the chess club, given that the student does not play in the orchestra.

Find the probability that a randomly selected cucumber is shorter than 30 cm.

Given that a randomly selected cucumber is shorter than 30 cm, find the probability that it is longer than 26 cm.

Only cucumbers longer than 25 cm are sold to a supermarket. Find the value of $k$ such that half of the cucumbers sold to the supermarket are longer than $k$ cm.

Consider the case $\mathrm{P}(F)=0.1$ and $a=0.8$. A component is tested once and fails. Find the probability that it is faulty.

Show that $\mathrm{O}(F\mid \text{fail})={\displaystyle \frac{a}{1-a}}\times \mathrm{O}(F)$.

Find a similar expression for $\mathrm{O}(F\mid \text{pass})$ in terms of $a$ and $\mathrm{O}(F)$.

Write down $\mathrm{P}(R_k\mid F)$ and $\mathrm{P}(R_k\mid F^{\mathrm{\prime }})$.

Hence show that $\mathrm{O}(F\mid R_k)={\left({\displaystyle \frac{a}{1-a}}\right)}^k\times \mathrm{O}(F)$.

A component is tested $m+n$ times. It fails $m$ of the tests and passes the other $n$, in some order. Let $S$ be the event that these results are obtained, in that order. Show that $\mathrm{O}(F\mid S)={\left({\displaystyle \frac{a}{1-a}}\right)}^{m-n}\times \mathrm{O}(F)$.

Explain briefly the significance of the result in part (e) for a component that fails exactly as many tests as it passes.

A component is tested 10 times. It fails 6 tests and passes 4. Find the probability that it is faulty.

A component is tested repeatedly and fails every test. Find the least number of tests after which the probability that it is faulty is greater than 0.999.

A faulty component is tested 10 times. Find the probability that, from its results, the probability that it is faulty is calculated to be greater than 0.5.