Practice section 2 of 7

A student has probability 0.1 of being late for school on any day. In 18 days, the probability of being late on fewer than r days must exceed 0.85. Find the least r.

A child wins each computer game with probability p. Out of 20 games, the probability of at least two wins is 0.06. Write an equation for p and solve it numerically.

A fixed number n of cars is observed and R counts those that are red. (i) State in context two assumptions for a binomial model. Assume now $R\sim B(n,p)$. (ii) For n=20 and p=0.15 find $P(4\le R<8)$. (iii) For n=20 and $P(R=0\text{ or }1)=0.2$, write an equation for p and find p numerically.

A box contains two red balls among seven balls. A ball is drawn randomly, its colour noted, then returned and mixed with the others. Repeat 50 times.

(i) Find the expected red count μ and standard deviation σ.

(ii) Find $P(\mu-\sigma<X<\mu+\sigma)$.

In a village with a large population, 80% of inhabitants have a particular eye disorder. (i) If 12 people wait to see the nurse, find the most likely number with the disorder. (ii) If 19 people wait, find the most likely number.

Five people are sampled from a large village. (i) X counts those who are underweight; state in context an assumption needed for a binomial model. (ii) The village proportion underweight is p, and the mode of X is 2. Show that $1/3<p<1/2$. (iii) Given instead $X\sim B(5,0.25)$, sketch its distribution.