Kirk and Les regularly play each other at darts. The probability that Kirk wins any game is 0.3 and the outcome of each game is independent of the outcome of every other game. a)Find the probability that in a match of 15 games Kirk wins
b)find the expected number of game in a match of 15 games which Kirk will win
Thanks in advance
This is a typical case of application of the binomial distribution, which requires:
1. probability of success (0.3) is known and remains constant throughout the experiment.
2. each trial is independent of the others.
3. there are exactly two possible outcomes, success or failure (in this case).
4. the number of trials is known and remains constant, i.e. independent of outcomes.
Here n=15, p=0.3 (for Kirk),
q=(1-p)=0.7
(a) For Kirk to win, he needs to win 8 matches or more, so
P(x≥8)=P(8)+P(9)+...P(15)
(b)
expected number of success,
E[x]=μ=np
a) To find the probability that Kirk wins an exact number of games in a match of 15, we can use the binomial probability formula. The binomial formula calculates the probability of a specific number of successes (in this case, Kirk wins) in a fixed number of independent trials (the 15 games).
The formula for the probability of exactly k successes in n trials, each with probability p of success, is given by:
P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))
In this case, we want to find the probability that Kirk wins exactly x games in a match of 15 games. So, we substitute the given values into the formula:
n = 15 (number of games in the match)
k = x (number of games Kirk wins in the match)
p = 0.3 (probability that Kirk wins any given game)
Using this formula, we plug in the numbers to find the probability of Kirk winning a specific number of games.
b) To find the expected number of games that Kirk will win in a match of 15 games, we can use the expected value formula for a binomial distribution.
The formula for the expected value (E) of the number of successes in n trials, each with probability p of success, is given by:
E(X) = n * p
In this case, we want to find the expected number of games that Kirk will win in a match of 15 games. So, we substitute the given values into the formula:
n = 15 (number of games in the match)
p = 0.3 (probability that Kirk wins any given game)
Using this formula, we calculate the expected number of games that Kirk will win in the match.
Remember that probability and expected value are theoretical calculations, and the actual outcome may vary in practice.