Jacob has four coins. One of the coins is biased so that, when it is tossed, the chance of getting a head is $\frac{7}{10}$. The other three coins are fair. Jacob tosses all four coins once. Let the random variable $X$ be the number of heads obtained. A probability table for $X$ is shown, with $x = 0,1,2,3,4$ and probabilities $P(X=0)=\frac{3}{80}$, $P(X=1)=a$, $P(X=2)=b$, $P(X=3)=c$, $P(X=4)=\frac{7}{80}$.
(a)[4]
Show that $a = \frac{1}{5}$ and find the values of $b$ and $c$.
(b)[1]
Calculate $\text{E}(X)$.
(c)[3]
Jacob tosses all four coins together $10$ times. Find the probability that exactly one head occurs on fewer than $3$ of those occasions.
(d)[2]
Find the probability that Jacob gets exactly one head for the first time on the $7$th or $8$th occasion that he throws the $4$ coins.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “An accurate unsimplified expression for $a=P(\text{1 head})$” …