Question

. Penalty Kicks Argentine soccer player Lionel Messi converts 78% of his penalty kicks. Suppose Messi takes six penalty kicks next season. Find the probability that the number he converts is exactly six

Answer 1

To find the probability that Lionel Messi converts exactly six penalty kicks, we can use the binomial probability formula.

Step-by-step explanation:

To find the probability that Lionel Messi converts exactly six penalty kicks, we can use the binomial probability formula:

P(X=k) = C(n,k) *
`p^(k)`*
`(1-p)^(n-k)`

where n is the number of trials (in this case, 6 penalty kicks), k is the number of successes (converting penalty kicks), p is the probability of success (78% or 0.78), and C(n,k) is the number of combinations of n objects taken k at a time. In this case, since we want exactly six converted penalty kicks, we plug in n = 6, k = 6, and p = 0.78.

P(X=6) = C(6,6) * 0.78⁶ * (1-0.78)⁽⁶⁻⁶⁾ = 1 * 0.78⁶ * 0.22⁰ = 0.78⁶ = 0.2997

Therefore, the probability that Lionel Messi converts exactly six penalty kicks is 0.2997 or 29.97%.

Answer 2

To find the probability that Lionel Messi converts all six penalty kicks, we can use the binomial probability formula. The probability that Messi converts all six penalty kicks is approximately 0.2051, or 20.51%.

Step-by-step explanation:

To find the probability that Lionel Messi converts all six penalty kicks, we can use the binomial probability formula.

Let's define a success as Messi converting a penalty kick. The probability of success is 0.78. Since each penalty kick is independent, we can use the binomial probability formula:

`P(X=k) = C(n, k) * p^k * (1-p)^(n-k)`

Where P(X=k) is the probability that Messi converts exactly k penalty kicks, n is the total number of penalty kicks he takes, k is the number of successful penalty kicks, p is the probability of success, and C(n, k) is the number of combinations of n items taken k at a time (given by n! / (k! * (n-k)!)).

Substituting the values into the formula, we have:

`P(X=6) = C(6, 6) * 0.78^6 * (1-0.78)^(6-6)`

`= 1 * 0.78^6 * 0.22^0`

= 0.2051114896

Therefore, the probability that Messi converts all six penalty kicks is approximately 0.2051, or 20.51%.