Calculating Binomial Probabilities
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.
What is [tex]$P(X=3)$[/tex] ? Enter your answer to 3 decimal places.
Recall: [tex]$P(X=k)=\binom{n}{k} p^k(1-p)^{n-k}$[/tex]
[tex]$P(X=3)= \square$[/tex]
Answer (1)
Use the binomial probability formula: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k .
Substitute n = 3 , k = 3 , and p = 0.4 into the formula.
Calculate each term: ( 3 3 ) = 1 , ( 0.4 ) 3 = 0.064 , and ( 0.6 ) 0 = 1 .
Multiply the terms to get the final probability: P ( X = 3 ) = 1 ∗ 0.064 ∗ 1 = 0.064 .
Explanation
Understand the problem We are given a binomial probability problem where Mrs. Chauvet rolls an unfair number cube 3 times. The probability of rolling a 6 is 40%, or 0.4. We want to find the probability of rolling exactly three 6s in those 3 trials.
State the formula We will use the binomial probability formula: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k , where:
n is the number of trials
k is the number of successes we want
p is the probability of success on a single trial
Plug in the values In our case, n = 3 , k = 3 , and p = 0.4 . Plugging these values into the formula, we get: P ( X = 3 ) = ( 3 3 ) ( 0.4 ) 3 ( 1 − 0.4 ) 3 − 3
Calculate each term Let's calculate each part:
( 3 3 ) = 1 (There is only one way to choose 3 successes from 3 trials)
( 0.4 ) 3 = 0.4 ∗ 0.4 ∗ 0.4 = 0.064
( 1 − 0.4 ) 3 − 3 = ( 0.6 ) 0 = 1 (Any number raised to the power of 0 is 1)
Multiply the terms Now, multiply these values together: P ( X = 3 ) = 1 ∗ 0.064 ∗ 1 = 0.064
State the answer Therefore, the probability of rolling exactly three 6s in 3 trials is 0.064.
Examples
Consider a basketball player who makes a free throw 40% of the time. If he takes 3 free throws, the probability he makes all 3 is calculated using the same binomial probability formula. This helps coaches and players understand the likelihood of achieving specific outcomes in repeated independent trials.
