A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. Find the probability of each value of $P$.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. Find the expected value of $P$.

A game involves spinning two spinners. One is numbered $1,2,3,4$. The other is numbered $2,2,4,4$. Each spinner is spun once and the number on each is recorded. Let $P$ be the product of the numbers on the spinners. A mathematician determines the amount of pocket money to give his son each week by getting him to play the game on Monday morning. If the son spins and the product is greater than $10$, then the boy gets $\pounds 10$. Otherwise, the boy gets $\pounds 5$. Find how much in total should the boy expect to get after $10$ weeks of playing the game.

Let $X$ be normally distributed with mean $75$ and standard deviation $5$.

Given that $P(X<65)=P(X>a)$, find the value of $a$.

Let $X$ be normally distributed with mean $75$ and standard deviation $5$.

Given that $P(65<X<a)=0.954$, Find $P(X>a).$

Data: $P(Z<-2)=P(Z>2)=0.023$

Three dice are thrown. If a $1$ or a $6$ is rolled somewhere on these three dice, you will be paid $\$1$, but if neither are rolled you will pay $\$5$. You play the game,

Find the probability that you will win $\$1$.

Three dice are thrown. If a $1$ or a $6$ is rolled somewhere on these three dice, you will be paid $\$1$, but if neither are rolled you will pay $\$5$. You play the game,

Complete the table shows the probability distribution of $X$, 'the number of dollars won in a game'.