Question

Based on the 2009 season, the Texas Rangers have a winning percentage of .533. Use the binomial model to find the probability that the Rangers win 4 of their next 5 games.

Answer 1

The probability that the Texas Rangers win 4 out of their next 5 games can be calculated using the binomial probability formula by plugging in the values for the number of games (n), number of wins (k), and the winning probability per game (p).

Step-by-step explanation:

The probability of the Texas Rangers winning 4 out of their next 5 games can be calculated using the binomial probability formula: P(X = k) = C(n, k) × p^k × (1 - p)^(n - k), where C(n, k) is the number of combinations of n things taken k at a time, p is the probability of a single game win, n is the number of games, and k is the number of wins.

For the Texas Rangers with a winning probability of .533, and wanting to win 4 out of 5 games, we have:

• n = 5 (number of games)
• k = 4 (number of wins)
• p = .533 (winning probability per game)

So the probability P(X = 4) will be:

P(X = 4) = C(5, 4) × .533^4 × (1 - .533)^1

After calculating the values, we find the probability for the Rangers to win exactly 4 out of 5 games.

Answer 2

0.128

Step-by-step explanation:

We know the probability for any event X is given by,

`P(X=x)=\binom{n}{x}* p^(n-x)* q^(x)`,

where p is the probability of success and q is the probability of failure.

Here, we are given that p = 0.533.

Since, we have that q = 1 - p

i.e. q = 1 - 0.533

i.e. q = 0.467

It is required to find the probability of 4 wins in the next 5 games i.e. P(X=4) when n = 5.

Substituting the values in the above formula, we get,

`P(X=4)=\binom{5}{4}* 0.533^(5-4)* 0.467^(4)`

i.e.
`P(X=4)=5 * 0.533 * 0.048`

i.e.
`P(X=4)=5 * 0.533 * 0.048`

i.e. i.e.
`P(X=4)=0.128`

Hence, the probability of 4 wins in the next 5 games is 0.128.