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]$P(X=1)$[/tex]? Enter your answer to 3 decimal places.
Recall: [tex]$P(X=k)=\binom{n}{k} p^k(1-p)^{n-k}$[/tex]
[tex]$P(X=1)=\binom{3}{1}(0.4)^1(0.6)^2=\square$[/tex]
Answer (1)
Use the binomial probability formula: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k .
Substitute n = 3 , k = 1 , and p = 0.4 into the formula: P ( X = 1 ) = ( 1 3 ) ( 0.4 ) 1 ( 0.6 ) 2 .
Calculate ( 1 3 ) = 3 , ( 0.4 ) 1 = 0.4 , and ( 0.6 ) 2 = 0.36 .
Multiply the values to get the final probability: P ( X = 1 ) = 3 × 0.4 × 0.36 = 0.432 .
Explanation
Problem Analysis We are given that Mrs. Chauvet has an unfair number cube that lands with 6 facing up 40% of the time. We want to find the probability that she rolls a 6 exactly once in 3 trials. We can use the binomial probability formula to solve this problem.
Binomial Probability Formula The binomial probability formula is given by: 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 to find the probability for
p is the probability of success on a single trial
( k n ) is the binomial coefficient, which can be calculated as k ! ( n − k )! n !
Identify Given Values In this problem, we have:
n = 3 (number of trials)
k = 1 (number of times we want to roll a 6)
p = 0.4 (probability of rolling a 6 on a single trial)
Substitute Values into Formula Now, we can plug these values into the binomial probability formula: P ( X = 1 ) = ( 1 3 ) ( 0.4 ) 1 ( 1 − 0.4 ) 3 − 1 P ( X = 1 ) = ( 1 3 ) ( 0.4 ) 1 ( 0.6 ) 2
Calculate Binomial Coefficient First, let's calculate the binomial coefficient: ( 1 3 ) = 1 ! ( 3 − 1 )! 3 ! = 1 ! 2 ! 3 ! = ( 1 ) ( 2 × 1 ) 3 × 2 × 1 = 3
Calculate Probabilities Next, let's calculate ( 0.4 ) 1 and ( 0.6 ) 2 :
( 0.4 ) 1 = 0.4 ( 0.6 ) 2 = 0.36
Calculate Final Probability Now, we can plug these values back into the formula: P ( X = 1 ) = 3 × 0.4 × 0.36 P ( X = 1 ) = 0.432
Final Answer Therefore, the probability of rolling a 6 exactly once in 3 trials is 0.432.
Examples
Binomial probabilities are useful in many real-world scenarios. For example, consider a basketball player who makes a free throw 70% of the time. We can use the binomial probability formula to calculate the probability that the player makes exactly 3 free throws out of 5 attempts. This type of calculation can help coaches and players understand the likelihood of different outcomes and make informed decisions. Another example is in quality control, where we can calculate the probability of finding a certain number of defective items in a sample from a production line.
