Worked solution
Convert the hit count into a score
Teams of 8 people enter a shooting competition. Each member of a team takes a shot at the target. The number of points scored by a team is obtained by where is the number of shots that hit the target. For a particular team, each member of the team independently has a probability of 0.75 of hitting the target.
(i) Find the probability that a randomly chosen team scores more than 6 points.
(ii) Find the number of points that a team is most likely to score.
CJC · Discussion · Question 7
Answer
(i) . (ii) points.
(i)
Let X count successful shots out of eight. Then . More than six points means 2X>6, hence X>3. Therefore (3 s.f.).
(ii)
The printed neighbouring probabilities are:
| x | P(X=x) |
|---|---|
| 5 | 0.20764 |
| 6 | 0.31146 |
| 7 | 0.26697 |
The modal hit count is six, giving a most likely score of 12 points. Independently, the binomial ratio rises to k=6 and then falls.
Common mistakes
The modal count is six hits; the question asks for the score, which is twice that.