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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.

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.

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.

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.