Question

1. Patients are known to arrive at a pharmacy randomly, with an average rate of four patients arriving per hour

1.1 What is a probability that exactly three patients will arrive at the pharmacy during a particular hour

1.2 What is the probability that only one patient will arrive at the pharmacy during a 30 minutes interval

1.3 What is the probability that seven patient will arrive at the pharmacy during a two hour interval

2. An aeroscape compagny has submitted bids on two separate government contracts A and B. the compagny feels that it has 50% chance of winning contract A and 40% chance of winning contract B. Futhermore, it belives that winning contract A is independent of winning contract B.

2.1 What is the probability that the compagny will win both contracts?

2.2 What is the probability that the compagny will win at least one of the contracts?

3. A compagny has two vacancies at the junior executive level. Ten people, consisting of seven men and three women, are eligible and equally qualified. The compagny has decided to draw two names at random from the list of those eligible.

3.1 What is the probability that both positions will be filled by women?

3.2 What is the probability that at least one of the positions will be filled by a woman?

3.3 What is the probability that both positions will be filled by a man?

3.4 From your answers, what conclusions can be reached about the vacancies?

Answer 1

1. Patients are known to arrive at a pharmacy randomly, with an average rate of four patients arriving per hour

Thiis is poisson event since it tells us the average rate. Poisson gives information for rate of event within a specified interval

The lambda = mean therefore = 4

This can be modified for eg: if we want to know for 2 hours then lambda = 4 * 2 = 8

1.1 What is a probability that exactly three patients will arrive at the pharmacy during a particular hour

This is for 1 hour so

P(X = 3) =

P(X = 3) = 0.1954

1.2 What is the probability that only one patient will arrive at the pharmacy during a 30 minutes interval

Here the interval is half hour so the average will reduce by half too

(4 /2)

P(X = 1) =

P(X = 1) = 0.2707

1.3 What is the probability that seven patient will arrive at the pharmacy during a two hour interval

Here the interval is 2 hour so the average will increase to double too

  (4 *2)

P(X = 7) =

P(X = 7) = 0.1396

2. An aeroscape compagny has submitted bids on two separate government contracts A and B. the compagny feels that it has 50% chance of winning contract A and 40% chance of winning contract B. Futhermore, it belives that winning contract A is independent of winning contract B.

Both contracts are independent of either so P(A and B) = P(A) * P(B). We are going to consider all events with respect to both A and B.

P(A') = 1- P(A) = 0.50 P(B') = 1 - 0.4 = 0.6

Where ' denotes the complement or losing the contract

2.1 What is the probability that the compagny will win both contracts?

Both A and B win

P(A and B) = P(A) * P(B)

=0.5 * 0.4

P(A and B) = 0.2

2.2 What is the probability that the compagny will win at least one of the contracts?

At least one win means either one win or both win

if one win then one will lose therefore either winning = P(A) * P(B') + P(A') * P(B) .........considering win or lose with both events

= 0.5 * 0.6 + 0.5 * 0.4

= 0.5

P(At least one win) = P(one wins) + P(both wins)

= 0.5 + 0.2 ....................(both wins from previous q)

P(at least one) = 0.7  

3. A compagny has two vacancies at the junior executive level. Ten people, consisting of seven men and three women, are eligible and equally qualified. The compagny has decided to draw two names at random from the list of those eligible.

For this we will use the combination function. Since here the order doesn't matter for eg choosing MW is same as WM given in both same man and woman are selected. Thw following is the formula

Here the ways to choose any two people out of 10 = 10C2 = 45

Prob = Desired outcomes / Total outcomes

Here the desired are the ones that we want specific selection like two men or two women or a man and woman. The denominator is any two. Therefore the denominator = 45.

3.1 What is the probability that both positions will be filled by women?

There are 3 women so both members by women = 3C2 = 3

Prob = 3 / 45

Probability = 0.067

3.2 What is the probability that at least one of the positions will be filled by a woman?

At least one means either man and women or both woman

One man and one women means 1 out of 7 men and 1 out of 3 women = 7C1 * 3C1 = 21

At least one women = 21 +3 = 24

Prob = 24 / 45

Ans: 0.533

3.3 What is the probability that both positions will be filled by a man?

Both by men means 2 out of 7 = 7C2 = 21

Prob = 21 / 45

Ans: 0.467

Both man means no women

P(no women) = 1- at least one women

= 1 - 0.533 - 0.467

3.4 From your answers, what conclusions can be reached about the vacancies?

There is higher chances for at least one man to be selected due to being higher in number.