Thuy rolls a number cube 7 times. Which expression represents the probability of rolling a 4 exactly 2 times?
[tex]P(k \text { successes }) = { }_n C_k p^k(1-p)^{n-k}[/tex]
[tex]{ }_n C_k =\frac{n!}{(n-k)!\cdot k!}[/tex]
A. [tex]{ }_7 C_5\left(\frac{1}{6}\right)^2\left(\frac{1}{6}\right)^5[/tex]
B. [tex]{ }_7 C_5\left(\frac{1}{6}\right)^5\left(\frac{5}{6}\right)^2[/tex]
C. [tex]{ }_7 C_2\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^5[/tex]
D. [tex]{ }_7 C_2\left(\frac{2}{6}\right)^2\left(\frac{4}{6}\right)^5[/tex]
Answer (1)
The problem asks for the probability of rolling a 4 exactly 2 times in 7 rolls of a number cube.
We use the binomial probability formula: P ( k ) = n C k p k ( 1 − p ) n − k .
Substituting the given values, we get 7 C 2 ( 6 1 ) 2 ( 6 5 ) 5 .
Therefore, the correct expression is 7 C 2 ( 6 1 ) 2 ( 6 5 ) 5 .
Explanation
Understand the problem and provided data We are given a problem where Thuy rolls a number cube 7 times and we want to find the probability of rolling a 4 exactly 2 times. We can use the binomial probability formula to solve this problem. The formula is given by: P ( k successes ) = n C k p k ( 1 − p ) n − k where n is the number of trials, k is the number of successes, p is the probability of success on a single trial, and n C k is the number of combinations of n items taken k at a time.
Identify the parameters In this problem, we have: n = 7 (number of rolls) k = 2 (number of times we want to roll a 4) p = 6 1 (probability of rolling a 4 on a single roll) 1 − p = 6 5 (probability of not rolling a 4 on a single roll)
Apply the binomial probability formula Now, we substitute these values into the binomial probability formula: P ( 2 successes ) = 7 C 2 ( 6 1 ) 2 ( 6 5 ) 7 − 2 = 7 C 2 ( 6 1 ) 2 ( 6 5 ) 5 We know that 7 C 2 = ( 7 − 2 )! ⋅ 2 ! 7 ! = 5 ! ⋅ 2 ! 7 ! = 2 × 1 7 × 6 = 21 .
State the final expression So, the expression for the probability of rolling a 4 exactly 2 times in 7 rolls is: 7 C 2 ( 6 1 ) 2 ( 6 5 ) 5 This matches the third option provided.
Examples
Consider a quality control scenario where a factory produces items, and each item has a probability of being defective. If we inspect a batch of items, we can use the binomial probability formula to calculate the probability of finding a certain number of defective items in the batch. For example, if we inspect 10 items and the probability of an item being defective is 5%, we can calculate the probability of finding exactly 2 defective items using the binomial probability formula. This helps in assessing the quality of the production process and making informed decisions about quality control measures.
