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.
$p(x) = \left \frac{n!}{x!(n - x)!} \
ight p^x q^{n - x}$
(1 point)
\bigcirc 12.4%
\bigcirc 18.8%
\bigcirc 23.7%
\bigcirc 50%

Explanation:

Step1: Identify parameters

Here, \( n = 5 \) (total games), \( x = 4 \) (wins), \( p = 0.533 \) (winning probability), \( q = 1 - p = 1 - 0.533 = 0.467 \).

Step2: Calculate combination

The combination formula is \( \frac{n!}{x!(n - x)!} \). For \( n = 5 \), \( x = 4 \):

$$ \frac{5!}{4!(5 - 4)!} = \frac{5!}{4!1!} = \frac{5\times4!}{4!×1} = 5 $$

Step3: Apply binomial formula

The binomial probability formula is \( P(x)=\frac{n!}{x!(n - x)!}p^{x}q^{n - x} \). Substitute the values:

$$ P(4)=5\times(0.533)^{4}\times(0.467)^{1} $$

First, calculate \( (0.533)^{4} \approx 0.533\times0.533\times0.533\times0.533 \approx 0.0819 \)
Then, \( (0.467)^{1}=0.467 \)
Multiply them with 5: \( 5\times0.0819\times0.467 \approx 5\times0.0383 \approx 0.1915 \approx 19.15\% \), which is closest to 18.8%.

Answer:

18.8% (the option with 18.8%)