Question

A restaurant has four bottles of a certain wine in stock. The wine steward does not know that two of these bottles (Bottles 1 and 2) are bad. Suppose that two bottles are ordered, and the wine steward selects two of the four bottles at random. Consider the random variable \(x=\) the number of good bottles among these two. a. One possible experimental outcome is (1,2) (Bottles 1 and 2 are selected) and another is (2,4) . List all possible outcomes. b. What is the probability of each outcome in Part (a)? c. The value of \(x\) for the (1,2) outcome is 0 (neither selected bottle is good), and \(x=1\) for the outcome (2,4) . Determine the \(x\) value for each possible outcome. Then use the probabilities in Part (b) to determine the probability distribution of \(x\). (Hint: See Example 6.5 )

Short answer

The possible outcomes from randomly selecting two bottles are: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) and (3, 4). The probability for each possible outcome is \(1/6\). The value of \(x\) for each outcome respectively is: 0, 1, 1, 1, 1 and 2. The probability distribution of \(x\) is as follows: \(x = 0\) with probability \(1/6\); \(x = 1\) with probability \(2/3\); \(x = 2\) with probability \(1/6\).

Step-by-step solution

Determining all outcomes

Find all the possible combinations of the 4 bottles taken 2 at a time. This would result in outcomes as follows: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) and (3, 4).

Calculate the probability for each outcome

Since there are 6 possible ways in which the steward can choose 2 bottles and assuming the steward is equally likely to choose any combination of bottles, the probability of selecting any one combination is \(1/6\). Therefore, the probability of each combination: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) and (3, 4) is \(1/6\).

Determine the x value for each outcome

For outcome (1,2), \(x = 0\) as both are bad bottles. For (1,3) and (1,4), \(x = 1\) as one of the bottles is good. Similar is the case for outcomes (2,3) and (2,4) where \(x = 1\). For outcome (3,4) however, \(x = 2\) as both are good bottles.

Determine the probability distribution of x

Outcome \(x = 0\) has 1 possibility (1,2) with combined probability of \(1/6\). Outcome \(x = 1\) has 4 possibilities ((1,3), (1,4), (2,3), (2,4)) with combined probability of \(4/6\) or \(2/3\). Outcome \(x = 2\) has 1 possibility (3,4) with probability of \(1/6\). Therefore, the probability distribution of \(x\) is as follows: \(x = 0\) with probability \(1/6\), \(x = 1\) with probability \(2/3\) and \(x = 2\) with probability \(1/6\).