A mattress store sells only king, queen and twin-size mattresses. Sales records at the store indicate that the number of queen-size mattresses sold is one-fourth the number of king and twin-size mattresses combined. Records also indicate that three times as many king-size mattresses are sold as twin-size mattresses.

Calculate the probability that the next mattress sold is either king or queen-size

• 0.12
• 0.15
• 0.80
• 0.85
• 0.95

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Letting t denote the relative frequency with which twin-sized mattresses are sold, we have that the relative frequency with which king-sized mattresses are sold is 3t and the relative frequency with which queen-sized mattresses are sold is (3t+t)/4, or t. Thus, t = 0.2 since t + 3t + t = 1. The probability we seek is 3t + t = 0.80.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Define Probabilities

Let's denote the probabilities of selling each type of mattress as follows:

• [math]P(K)[/math]: Probability of selling a king-size mattress.
• [math]P(Q)[/math]: Probability of selling a queen-size mattress.
• [math]P(T)[/math]: Probability of selling a twin-size mattress.

Step 2: Formulate Equations from Problem Statement

Based on the sales records, we can establish the following relationships:

1. "The number of queen-size mattresses sold is one-fourth the number of king and twin-size mattresses combined."
   [[math]]P(Q) = �rac{1}{4}(P(K) + P(T))[[/math]]
2. "Three times as many king-size mattresses are sold as twin-size mattresses."
   [[math]]P(K) = 3 P(T)[[/math]]
3. The sum of all probabilities must equal 1 (as these are the only types of mattresses sold):
   [[math]]P(K) + P(Q) + P(T) = 1[[/math]]

Step 3: Solve for Individual Probabilities

To simplify the calculation, let's express all probabilities in terms of [math]P(T)[/math]. Let [math]P(T) = t[/math]. From the second condition:

[[math]]P(K) = 3t[[/math]]

Now, substitute [math]P(K) = 3t[/math] and [math]P(T) = t[/math] into the first condition:

[[math]]P(Q) = �rac{1}{4}(3t + t)[[/math]]

[[math]]P(Q) = �rac{1}{4}(4t)[[/math]]

[[math]]P(Q) = t[[/math]]

Next, substitute these expressions for [math]P(K)[/math], [math]P(Q)[/math], and [math]P(T)[/math] into the third condition (the sum of probabilities equals 1):

[[math]]3t + t + t = 1[[/math]]

[[math]]5t = 1[[/math]]

[[math]]t = �rac{1}{5} = 0.2[[/math]]

Thus, the individual probabilities are:

• [math]P(T) = 0.2[/math]
• [math]P(K) = 3 imes 0.2 = 0.6[/math]
• [math]P(Q) = 0.2[/math]

Step 4: Calculate the Desired Probability

We need to calculate the probability that the next mattress sold is either king or queen-size. This is the sum of their individual probabilities:

[[math]]P(K text{ or } Q) = P(K) + P(Q)[[/math]]

[[math]]P(K text{ or } Q) = 0.6 + 0.2[[/math]]

[[math]]P(K text{ or } Q) = 0.8[[/math]]

The probability that the next mattress sold is either king or queen-size is 0.80.

Key Insights

• When dealing with mutually exclusive and exhaustive events (like different types of mattresses sold), their probabilities must sum to 1.
• Translating word problems into algebraic equations is a crucial first step in solving probability and ratio problems.
• Expressing all unknown probabilities in terms of a single variable (e.g., [math]t[/math] for [math]P(T)[/math]) can simplify a system of equations.
• The probability of either one event OR another (mutually exclusive) occurring is the sum of their individual probabilities.

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