Practise constructing discrete probability distributions, calculating expectation and variance and selecting binomial or geometric models with parameters tied to the experiment.
An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.
On another occasion, this die is thrown 3 times. The random variable X is the number of times that a 1 or a 6 is obtained.
Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6.
The first success is on throw 3, 4, or 5:
(31)(32)2+(31)(32)3+(31)(32)4.=274+818+24316.=24376=0.313…
On another occasion, this die is thrown 3 times. The random variable X is the number of times that a 1 or a 6 is obtained.
Draw up the probability distribution table for X.
| (X) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| (P(X)) | (278) | (2712) | (276) | (271) |
Find E(X).
E(X)=0(278)+1(2712)+2(276)+3(271).E(X)=2712+2712+273=1.
The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On 10 % of days less than 16 kg of grapes are sold.
The mass of grapes sold on any day is independent of the mass sold on any other day.
12 days are chosen at random.
Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
Let X∼Bin(12,0.1). Then
P(X<3)=1−[(012)(0.1)0(0.9)12+(112)(0.1)(0.9)11+(212)(0.1)2(0.9)10]=0.111.