Calculating Binomial Probabilities
Leigh plays a game at a fundraiser for which there is a probability of 0.2 that she wins. She plans to play this game times. What is the probability that she wins exactly 1 time? Enter your answer to 4 decimal places.
[tex]\begin{array}{l}
P(X=k)=\binom{n}{k} p^k(1-p)^{n-k} \\
P(X=1)=\square
\end{array}[/tex]
Answer (1)
Identify the values: n = 7 , k = 1 , p = 0.2 .
Apply the binomial probability formula: P ( X = 1 ) = ( 1 7 ) ( 0.2 ) 1 ( 0.8 ) 6 .
Calculate the binomial coefficient and probabilities: ( 1 7 ) = 7 , ( 0.2 ) 1 = 0.2 , ( 0.8 ) 6 = 0.262144 .
Calculate the final probability: P ( X = 1 ) = 7 \tims 0.2 \tims 0.262144 = 0.3670 .
Explanation
Understand the problem and provided data We are given a binomial probability problem where Leigh plays a game 7 times, and the probability of winning each game is 0.2. We want to find the probability that she wins exactly 1 time. The formula for binomial probability is: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k where:
n is the number of trials (games played)
k is the number of successful trials (games won)
p is the probability of success on a single trial (probability of winning a game)
( k n ) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.
Identify the values for n, k, and p In this problem, we have:
n = 7 (Leigh plays the game 7 times)
k = 1 (We want to find the probability of winning exactly 1 time)
p = 0.2 (The probability of winning a single game is 0.2)
We can plug these values into the binomial probability formula: P ( X = 1 ) = ( 1 7 ) ( 0.2 ) 1 ( 1 − 0.2 ) 7 − 1
Calculate the binomial coefficient First, let's calculate the binomial coefficient ( 1 7 ) :
( 1 7 ) = 1 ! ( 7 − 1 )! 7 ! = 1 ! 6 ! 7 ! = ( 1 ) × ( 6 × 5 × 4 × 3 × 2 × 1 ) 7 × 6 × 5 × 4 × 3 × 2 × 1 = 1 7 = 7
Calculate p^k Next, let's calculate ( 0.2 ) 1 :
( 0.2 ) 1 = 0.2
Calculate (1-p)^(n-k) Now, let's calculate ( 1 − 0.2 ) 7 − 1 :
( 1 − 0.2 ) 7 − 1 = ( 0.8 ) 6 We can calculate ( 0.8 ) 6 :
( 0.8 ) 6 = 0.262144
Calculate the final probability Now, we can substitute these values back into the binomial probability formula: P ( X = 1 ) = 7 × 0.2 × 0.262144 = 1.4 × 0.262144 = 0.3670016 Rounding to 4 decimal places, we get: P ( X = 1 ) ≈ 0.3670
State the final answer Therefore, the probability that Leigh wins exactly 1 time out of 7 games is approximately 0.3670.
Examples
Binomial probabilities are useful in many real-world scenarios. For example, consider a quality control process in a factory where a certain percentage of products are defective. If you take a sample of products, you can use the binomial probability formula to calculate the probability of finding a specific number of defective items in your sample. This helps in assessing the effectiveness of the manufacturing process and making informed decisions about quality control measures. Another example is in medical research, where binomial probabilities can be used to determine the likelihood of a certain number of patients experiencing side effects from a new drug, given the probability of a single patient experiencing the side effect.
