Probability Theory By Examples

Example 1

The Ludlow Wildcats baseball team, a minor league team in the Cleveland Indians organization, plays 70 percent of their games at night and 30 percent during the day. The team wins 50 percent of their night games and 90 percent of their day games. According to today's newspaper, they won yesterday. What is the probability the game was played at night?

% of games played at night = 70% % of games played during day = 30% % of night games won =50% % of day games won= 90%

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Probability of winning = Probability of winning at night + Probability of winning during day = % of games played at night x % of night games won + % of games played during day x % of day games won = 70% x 50% + 30% x 90% = 0. 35 + 0. 27 = 0. 62 Probability that the game was played during night given that the game was won = Probability of winning at night / Probability of winning = 0. 35 / 0. 62 = 35/62 Answer: Probability = 35/62 This can be understood in a different way

Let the number of games played be 100 Out of these 100 games, 70 games were played at night and 30 during day Out of 70 games played at night no of games won = 50% x 70 = 35 games and the number of games lost = 50% x 70 =35 Out of 30 games played during day, no of games won = 90% x 30 = 27 games and the number of games lost = 10% x 30 = 3 Thus total games won = 35 + 27 = 62 (Total games lost = 35 + 3 =38, but this is not required for calculation) Thus out of 62 games won , 35 were won at night

Thus probability that the game was played at night, given that the game was won = 35/62

Example 2

With each purchase of a large pizza at Tony's Pizza, the customer receives a coupon that can be scratched to see if a prize will be awarded. The odds of winning a free soft drink are 1 in 10, and the odds of winning a free large pizza are 1 in 50. You plan to eat lunch tomorrow at Tony's.

What is the probability:

• That you will win either a large pizza or a soft drink
• That you will not win a prize?
• That you will not win a prize on three consecutive visits to Tony's
• That you will win at least one prize on one of your next three visits to Tony's

We have to convert odds into probability Probability = odds / (1+ odds) Odds of winning a free soft drink are 1 in 10 Therefore, probability of winning a free soft drink = (1/10) / (1 + 1/10) = 1/11 Odds of winning a free large pizza are 1 in 50 Therefore, probability of winning a free large pizza = (1/50) / (1 + 1/50) = 1/51

That you will win either a large pizza or a soft drink The events winning a pizza and winning a soft drink are mutually exclusive (since you can either win a pizza or you can win a soft drink but not both at the same time as you have only one coupon ) Probability of winning either a large pizza or a soft drink = Probability of winning a large pizza + Probability of winning a soft drink = 1/51 + 1/11 = 62 /561 = 0. 11 or 11%

Probability of not winning a prize = Probability of winning a prize = 1- 62/561 = 499/561 = 0. 9 or 89% 3. That you will not win a prize on three consecutive visits to Tony's Since the events of winning / not winning on consecutive visits are independent events we will multiply the probabilities Probability of not winning a prize on three consecutive visits = Probability of not winning on first visit x Probability of not winning on second visit x Probability of not winning on third visit = (499 /561) x (499 / 561) x (499 / 561) = (499/561) ^3 = 0. 70 or 70%

Probability of winning at least once = 1- probability of not winning even once = 1- (499/561) ^3 = 0. 30 or 30%

Example 3

There are four people being considered for the position of chief executive officer of Dalton Enterprises. Three of the applicants are over 60 years of age. Two are female, of which only one is over 60.

1. What is the probability that a candidate is over 60 and female
2. Given that the candidate is male, what is the probability he is less than 60
3. Given that the person is over 60, what is the probability the person is female

Out of 4 applicants Male = 2 (both over 60) Female = 2 (1 over 60, 1 less than 60) 1) What is the probability that a candidate is over 60 and female Out of 4 candidates only 1 is both female and over 60 Therefore probability = ? = 0. 25 or 25% 2. Given that the candidate is male, what is the probability he is less than 60 Both male candidates are over 60 therefore probability = 0 3 Given that the person is over 60, what is the probability the person is female There are 3 persons over 60 out of which 1 is female. Therefore, probability = 1/3.

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