The probability that player A will win any game is 25%, the probability that player B will win is 25% and the probability that player C will win is 50%. If they play 10 games, what is the probability that player A will win 2 games, player B will win 3 games and player C will win 5 games?

1. The probability that player A will win any game is
25%, the probability that player B will win is 25%
and the probability that player C will win is 50%.
If they play 10 games, what is the probability that
player A will win 2 games, player B will win 3
games and player C will win 5 games?
NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS
https://www.slideshare.net/NadeemUddin17
https://nadeemstats.wordpress.com/listofbooks/

2. Multinomial experiment
If each trial have more than two outcomes than the binomial experiment becomes a
multinomial experiment.
Multinomial distribution
If a given trial can result in the k outcomes 𝐸1, 𝐸2, 𝐸3, … , 𝐸𝑘, with probability
𝑝1, 𝑝2, 𝑝3, … , 𝑝𝑘, then the probability distribution of the random variables
𝑋1, 𝑋2, 𝑋3, … , 𝑋𝑘, representing the number of occurrences for 𝐸1, 𝐸2, 𝐸3, … , 𝐸𝑘 in “n”
independent trials, is
𝑃 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑘; 𝑝1, 𝑝2, 𝑝3, … , 𝑝𝑘, 𝑛 =
𝑛!
𝑥1! 𝑥2! 𝑥3! … … 𝑥𝑘!
𝑝1
𝑥1
𝑝2
𝑥2
𝑝3
𝑥3
… … 𝑝𝑘
𝑥𝑘
Where 𝑖=1
𝑘
𝑥𝑖 = 𝑛

3. Example-4
The probability that player A will win any game is 25%, the probability that player B
will win is 25% and the probability that player C will win is 50%. If they play 10
games, what is the probability that player A will win 2 games, player B will win 3
games and player C will win 5 games?
Solution
𝑛 = 10 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
𝑥1 = 2 𝑔𝑎𝑚𝑒𝑠
𝑥2 = 3 𝑔𝑎𝑚𝑒𝑠
𝑥3 = 5 𝑔𝑎𝑚𝑒𝑠

4. 𝑝1 = 𝑃 𝐴 𝑤𝑖𝑛𝑠 = 0.25
𝑝2 = 𝑃 𝐵 𝑤𝑖𝑛𝑠 = 0.25
𝑝3 = 𝑃 𝐶 𝑤𝑖𝑛𝑠 = 0.50
𝑃 2,3,5,0.25,0.25,0.50,10 =
10!
2! 3! 5!
0.25 2 0.25 3 0.50 5
𝑃 2,3,5,0.25,0.25,0.50,10 = 0.0769