Dean Chalmers and Julian Gilbey Solutions for Chapter: Probability Distributions, Exercise 5: EXERCISE 6B

A company offers a $\$1000$ cash loan to anyone earning a monthly salary of at least $\$2000.$ To secure the loan, the borrower signs a contract with a promise to repay the $\$1000$ plus a fixed fee before $3$ months have elapsed. Failure to do this gives the company a legal right to take $\$1540$ from the borrower's next salary before returning any amount that has been repaid.

From past experience, the company predicts that $70\%$ of borrowers succeed in repaying the loan plus the fixed fee before $3$ months have elapsed.

Calculate the fixed fee that ensures the company an expected $40\%$ profit from each $\$1000$ loan.

When a scout group of $8$ juniors and $12$ seniors meets on a Monday evening, one scout is randomly selected to hoist a flag. Let the variable $X$ represent the number of juniors selected over $n$ consecutive Monday evenings.

By drawing up the probability distribution table for $X,$ or otherwise, show that $E\left(X\right)=1.2$ when $n=3.$

When a scout group of $8$ juniors and $12$ seniors meets on a Monday evening, one scout is randomly selected to hoist a flag. Let the variable $X$ represent the number of juniors selected over $n$ consecutive Monday evenings.

Find the number of Monday evenings over which $14$ juniors are expected to be selected to hoist the flag.

An ordinary fair die is rolled. If the die shows an odd number then $S,$ the score awarded, is equal to that number. If the die shows an even number, then the die is rolled again. If on the second roll it shows an odd number, then that is the score awarded. If the die shows an even number on the second roll, the score awarded is equal to half of that even number.

Find $P\left[S>E\left(S\right)\right]$

An ordinary fair die is rolled. If the die shows an odd number then $S,$ the score awarded, is equal to that number. If the die shows an even number, then the die is rolled again. If on the second roll it shows an odd number, then that is the score awarded. If the die shows an even number on the second roll, the score awarded is equal to half of that even number.

Calculate the exact value of Var $\left(S\right)$

A fair $4$ -sided spinner with sides labelled $A, A, B, B$ is spun four times.

Show that there are six equally likely ways to obtain exactly two $Bs$ with the four spins.

A fair $4$ -sided spinner with sides labelled $A, A, B, B$ is spun four times.

By drawing up the probability distribution table for $X,$ the number of times the spinner comes to rest on $B,$ find the value of $\frac{Var\left(X\right)}{E\left(X\right)}.$

A fair $4$ -sided spinner with sides labelled $A, B, B, B$ is spun four times.

What, in the context of this question, does the value found in part $b$ represent?