A survey of benefits for 254 corporate executives (Business Week, October
24, 1994) showed that 155 executives were provided mobile phones, 152 were
provided club memberships, and 110 were provided both mobile phones and
club memberships as perks associated with their position.
(a) Let M be the event of having a mobile phone and C be the event of having a
club membership. Find the following probabilities: P (M ), P (C ), and
P (M ∩ C ).
(b) Use the probabilities in part (a) to compute the probability that a corporate
executive has at least one of the two perks.
(c) What is the probability that a corporate executive does not have either
of these perks.
15
(d) Organize the probabilistic information into a probability tree whose first
stage represents ownership of a mobile phone, and whose second stage
represents enrollment in club membership.
(e) Determine the probability that a corporate executive does not have club
membership given that that executive has a mobile phone.
(f ) Determine the probability that a corporate executive does not have a
mobile phone given that that executive does not have club membership.

(a) P(M) = 155/254

    P(C) = 76/127

    P(M ∩ C) = 55/127

(b) P(M U C) = 197/254

(c) P(Neither of the perks) = 57/254

(d) Probability tree drawn.

(e) P(C'|M) = 9/31

(f) P(M'|C') = 57/102

Step-by-step explanation:

The question states that:

Total executives = 254

Executives with mobile phones = 155

Executives with club memberships = 152

Executives with both mobile phones and club memberships = 110

(a) P(M) = No. of executives with mobile phones/Total no. of executives

            = 155/254

    P(M) = 155/254

P(C) = No. of executives with club memberships/Total no. of executives

       = 152/254

P(C) = 76/127

P(M ∩ C) = No. of executives with both mobile phones and club memberships/Total no. of executives

               = 110/254

P(M ∩ C) = 55/127

(b) We are asked to find the probability that a corporate has at least one of the two perks i.e. either they have a mobile phone or a club membership which means we need to find P(M U C).

P(M U C) = P(M) + P(C) - P(M ∩ C)

              = 155/254 + 152/254 - 110/254

P(M U C) = 197/254

(c) The probability that a corporate executive does not have either of these perks can be calculated by subtracting the probability that a corporate executive has at least one of these perks from the total probability (i.e. 1). So,

P(Neither of the perks) = 1 - P (M U C)

                = 1 - 197/254

P(Neither of the perks) = 57/254

(d) Probability tree can be drawn in two stages where the first stage represents the ownership of mobile phone and the second stage represents the ownership of club membership.

M = having a mobile phone

M' = not having a mobile phone

C = having a club membership

C' = not having a club membership

I have drawn the probability tree and attached it as an image.

(e) We will use the conditional probability formula here to calculate the probability that a corporate executive does not have club  membership given that that executive has a mobile phone

P(C'|M) = P(C' ∩ M) / P(M)

P(C' ∩ M) is the number of executives who do not have a club membership but only have a mobile phone. We can calculate the no. of executives with only mobile phones as:

Executives with mobile phones - Executives with both mobile phones and club memberships

= 155 - 110 = 45 executives with only mobile phones

So, P(C' ∩ M) = 45/254

P(C'|M) = (45/254)/(155/254)

P(C'|M) = 9/31

(f) We will again use the conditional probability formula here. We need P(M'|C'). So,

P(M'|C') = P(M' ∩ C')/(P(C')

P(M' ∩ C') represents the number of people who do not have a mobile phone nor a club membership. i.e. the number of corporate executives who have neither of these perks. We calculated this probability in part (c).

P(C') is the number of people who do not have a club membership. These include the number of people who have only a mobile phone and the people who have neither of these things. So,

P(C') = P(C' ∩ M) + P(M' U C')

        = 45/254 + 57/254

P(C') = 102/254

So, P(M'|C') = P(M' ∩ C')/(P(C')

                   = (57/254)/(102/254)

      P(M'|C') = 57/102