An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string of the sort oee (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll).For each outcome, let N be the random variable counting the number of odd rolls in each outcome. For example, if the outcome is oeo, then =Noeo2. Suppose that the random variable X is defined in terms of N as follows: =X−2N2−4N3. The values of X are given in the table below.Outcome ooo eoe eeo oeo ooe eee eoo oeeValue of X 3 −5 −5 −3 −3 −3 −3 −5Calculate the probabilities P=Xx of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row.Value x of X P=Xx
Question
An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string of the sort oee (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll).For each outcome, let N be the random variable counting the number of odd rolls in each outcome. For example, if the outcome is oeo, then =Noeo2. Suppose that the random variable X is defined in terms of N as follows: =X−2N2−4N3. The values of X are given in the table below.Outcome ooo eoe eeo oeo ooe eee eoo oeeValue of X 3 −5 −5 −3 −3 −3 −3 −5Calculate the probabilities P=Xx of the probability distribution of X. First, fill in the first row with the values of X. Then fill in the appropriate probabilities in the second row.Value x of X P=Xx
Solution
The first step is to understand the possible outcomes when a die is rolled. Since a die has 6 faces, each with an equal probability of being rolled, the probability of rolling an odd number (1, 3, or 5) or an even number (2, 4, or 6) is 1/2.
Next, we need to understand the possible outcomes when a die is rolled three times. There are 2^3 = 8 possible outcomes, which are listed in the table in the question.
Now, we need to calculate the value of X for each outcome. The formula for X is given as X = -2N^2 - 4N + 3, where N is the number of odd rolls in each outcome. The values of X for each outcome are already given in the table.
Finally, we need to calculate the probabilities P(X=x) for each value of X. Since each outcome is equally likely, the probability of each outcome is 1/8.
The values of X are -5, -3, and 3.
- The value -5 occurs for outcomes eoe, eeo, and oee, so P(X=-5) = 3/8.
- The value -3 occurs for outcomes oeo, ooe, eee, and eoo, so P(X=-3) = 4/8 = 1/2.
- The value 3 occurs for outcome ooo, so P(X=3) = 1/8.
So, the probability distribution of X is:
Value x of X: -5 -3 3 P(X=x): 3/8 1/2 1/8
Similar Questions
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