Answer to Question #188558 in Statistics and Probability for LE8

Question

Evelynn challenged Ahri and Kai'Sa to play a series of matches of darts. (with Akali eating her ramen and Seraphine being excited).

Here are the probabilities of each woman winning:

Kai'Sa: 50%; Evelynn: 30%; Ahri: 20%

If they play 6 games, Seraphine calculated that the probability of Ahri will win 1 game, Evelynn will win 2 games, and Kai'Sa will win 3 games is around Blank 1. (The probability is in decimal format with 4 decimal places.)

Akali is thoroughly impressed with Seraphine's arithmetic capabilities and decides to cook her some ramen as well.

Answer 1

By condition,

${p_1} = 0.5,\,\,{p_2} = 0.3,\,\,{p_3} = 0.2$

Using the polynomial scheme, we find  the probability of Ahri will win 1 game, Evelynn will win 2 games, and Kai'Sa will win 3 games:

$P(3,2,1) = \frac{{6!}}{{3!2!1!}}p_1^3p_2^2{p_1} = \frac{{4 \cdot 5 \cdot 6}}{2} \cdot {0.5^3} \cdot {0.3^2} \cdot 0.2 = 0.135$

Answer: P=0.135