Mrs. Chauvet has an unfair number cube that lands with 6 facing up [tex]$40 \%$[/tex] of the time. Let [tex]$X=$[/tex] the number of times that she rolls a 6 among 3 trials. Here is a binomial probability table for this setting.
| [tex]$X$[/tex] | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| [tex]$P(X)$[/tex] | [tex]$1(0.6)^3$[/tex] | [tex]$3(0.4)(0.6)^2$[/tex] | [tex]$3(0.4)^2(0.6)$[/tex] | [tex]$1(0.4)^3$[/tex] |
Complete this statement:
[tex]$X=1$[/tex] means that we roll exactly $\square$ 6 in the 3 trials. That means we have $\square$ success and failures.
Answer (1)
X = 1 means we roll exactly 1 six in 3 trials.
Rolling a 6 is considered a success, so we have 1 success.
With 3 trials and 1 success, we have 3 − 1 = 2 failures.
Therefore, we have 1 success and 2 failures: 1 success and 2 failures .
Explanation
Understand the problem The problem states that Mrs. Chauvet has an unfair number cube that lands with 6 facing up 40% of the time. We are given that X represents the number of times she rolls a 6 among 3 trials. We need to complete the statement: X = 1 means that we roll exactly □ 6 in the 3 trials. That means we have □ success and failures.
Determine the number of sixes Since X represents the number of times she rolls a 6, X = 1 means that she rolls exactly 1 six in the 3 trials.
Determine the number of successes In this context, rolling a 6 is considered a success. Therefore, if she rolls exactly 1 six, she has 1 success.
Determine the number of failures Since there are 3 trials and 1 success, the number of failures is the total number of trials minus the number of successes, which is 3 − 1 = 2 . Therefore, she has 2 failures.
Final Answer Therefore, X = 1 means that we roll exactly 1 six in the 3 trials. That means we have 1 success and 2 failures.
Examples
Consider a quality control process where you inspect 3 items from a production line, and you define 'success' as finding a defective item. If X = 1 , it means you found exactly one defective item in the 3 inspected. This helps assess the defect rate and the overall quality of the production.
