Question

The three most popular options on a certain type of newcar are a built-in GPS (A), a sunroof (B), and an automatictransmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request Aor B,77% request Aor C, 80% request Bor C, and 85% request Aor Bor C, determine the probabilities of the following events. (Hint:“Aor B” is the event that at leastone of the two options is requested; try drawing a Venn

diagram and labeling all regions.)

a. The next purchaser will request at least one of thethree options.

b. The next purchaser will select none of the three options.

c. The next purchaser will request only an automatictransmission and not either of the other two options.

d. The next purchaser will select exactly one of thesethree options.

Short answer

a.The probability that the next purchaser will request at least one of the three options is 0.85.

b.The probability that the next purchaser will select none of the three optionsis 0.15.

c.The probability the next purchaser will request only an automatic transmission and not either of the other two optionsis 0.22.

d.The probability that the next purchaser will select exactly one of these three options is 0.35.

Step-by-step solution

Given information

The number of options on a certain type of new car is 3; they are,

A: built-in GPS

B: a sunroof

C: an automatic transmission

The probability that all purchasers request A is\(P\left( A \right) = \)0.40.

The probability that all purchasers request B is\(P\left( B \right) = \)0.55.

The probability that all purchasers request C is\(P\left( C \right) = \)0.70.

The probability that all purchasers request A or B is\(P\left( {A \cup B} \right) = \)0.63.

The probability that all purchasers request A or C is\(P\left( {A \cup C} \right) = \)0.77

The probability that all purchasers request B or C is\(P\left( {B \cup C} \right) = \)0.80.

The probability that all purchasers request A or B or C is \(P\left( {A \cup B \cup C} \right) = \) 0.85.

Compute the probability

a.

From the provided data, theprobability that the next purchaser will request at least one of the three options is mathematically expressed as \(P\left( {A \cup B \cup C} \right) = \) 0.85.

Therefore, the probability that the next purchaser will request at least one of the three options is 0.85.

b.

The probability that the next purchaser will select none of the three optionsis computed as,

\(\begin{aligned}P\left( {A \cup B \cup C} \right)' &= 1 - P\left( {A \cup B \cup C} \right)\\ &= 1 - 0.85\\ &= 0.15\end{aligned}\)

Therefore, the probability that the next purchaser will select none of the three optionsis 0.15.

Compute the probabilities for the venn diagram

The probabilities required to make further computations are as follows,

\(\begin{aligned}P\left( {A \cap B} \right) &= P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right)\\ &= 0.40 + 0.55 - 0.63\\ &= 0.32\end{aligned}\)

\(\begin{aligned}P\left( {A \cap C} \right) &= P\left( A \right) + P\left( C \right) - P\left( {A \cup C} \right)\\ &= 0.40 + 0.70 - 0.77\\ &= 0.33\end{aligned}\)

\(\begin{aligned}P\left( {B \cap C} \right) &= P\left( B \right) + P\left( C \right) - P\left( {B \cup C} \right)\\ &= 0.55 + 0.70 - 0.80\\ &= 0.45\end{aligned}\)

\(\begin{aligned}P\left( {A \cap B \cap C} \right) &= P\left( {A \cup B \cup C} \right) - P\left( A \right) - P\left( B \right) - P\left( C \right) + P\left( {A \cap C} \right) + P\left( {A \cap C} \right) + P\left( {B \cap C} \right)\\ &= 0.85 - 0.40 - 0.55 - 0.70 + 0.32 + 0.33 + 0.45\\ &= 0.30\end{aligned}\)

Using the above probabilities that Venn diagram is represented as,

Note, the probabilities for intersection of two sets is further broken down using the intersection of all sets.

Compute the required probabilities

c.

The probability that the next purchaser will request only an automatic transmission and not either of the other two optionsis computed as,

\(\begin{aligned}P\left( {A' \cap B' \cap C} \right) &= P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right) + P\left( {A \cap B \cap C} \right)\\ &= 0.70 - 0.33 - 0.45 + 0.30\\ &= 0.22\end{aligned}\)

Therefore, the probability the next purchaser will request only an automatic transmission and not either of the other two optionsis 0.22.

d.

The probability that the next purchaser will request only built-in GPS and not either of the other two optionsis computed as,

\(\begin{aligned}P\left( {A \cap B' \cap C'} \right) &= P\left( A \right) - P\left( {A \cap C} \right) - P\left( {A \cap B} \right) + P\left( {A \cap B \cap C} \right)\\ &= 0.40 - 0.33 - 0.32 + 0.30\\ &= 0.05\end{aligned}\)

The probability that the next purchaser will request only sunroof and not either of the other two optionsis computed as,

\(\begin{aligned}P\left( {A' \cap B \cap C'} \right) &= P\left( B \right) - P\left( {B \cap C} \right) - P\left( {A \cap B} \right) + P\left( {A \cap B \cap C} \right)\\ &= 0.55 - 0.45 - 0.32 + 0.30\\ &= 0.08\end{aligned}\)

The probability that the next purchaser will select exactly one of these three options is computed as,

\(\begin{aligned}P\left( {A' \cap B' \cap C} \right) + P\left( {A' \cap B \cap C'} \right) + P\left( {A \cap B' \cap C'} \right) &= 0.22 + 0.05 + 0.08\\ &= 0.35\end{aligned}\)

Therefore, the probability that the next purchaser will select exactly one of these three options is 0.35.