Probabilities
Rule 1: Relative Frequency Approximation of Probability
P(A) = number of times A occured / number of times trial was repeated
Rule 2: Classical APproach to Probability (Equally likely outcomes)
P(A) = number of ways A can occur / number of different simple events
Rule 3: Subjective Probabilites
P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.
QUESTION Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume the boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child.
SOLUTION The biggest obstacle here is correctly identifying the sample space. It involves more than working only with the numbers 2 and 3 that were given in the statement of the problem. The sample space consists of 8 different ways that 3 children can occur. 2^n = 2^3 = 8. Those 8 outcomes are equally likely, so we use Rule 2. Of those 8 different possible outcomes, 3 correspond to exactly 2 boys, so P(2 boys in 3 births) = 3/8 = .375
Complementary Events
The completement of event A, denoted as Ā, consists of all outcomes in which event A does not occur.
Odds
The actual odds against event A occuring are the ratio P(Ā)/P(A), usually expressed in the form of a:b (or "a to b"), where a and b are integers having no common factors.
The actual odds in favor of event A are the reciprocal of the actual odds against that event. If the odds against A are a:b, then the odds in favor of A are b:a.
The payoff odds against event A represent the ratio of net profit (if you win) to the amount bet.
payoff odds against event A = (net profit):(amount bet)
QUESTION If you bet $5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino as 35:1.
a. Find the actual odds against the outcome of 13.
b. How much net profit would you make if you win by betting on 13?
c. If the casino were operating just for the fun of it, and the payoff odds were changed to match the actual odds against 13, how much would you win if the outcome were 13?
SOLUTION
a. With P(13)= 1/38 and P(not 13)= 37/38, we get actual odds against 13 = P(not 13)/P(13) = (37/38)/(1/38) = 37/1 or 37:1
b. Because the payoff odds against 13 are 35:1,
we have 35:1 = (net profit):(amount bet)
so that there is a $35 profit for each $1 bet. For a $5 bet, the net profit is $175. The winning bettor would collect $175 plus the original $5 bet. That is, the winning bettor of $5 would receive the $5 bet plus another $175. The total amount returned would be $180, for a net profit of $175.
c. If the casino were operating for fun and not for profit, the payoff odds would be equal to the actual odds against the outcome of 13. If the payoff odds were changed from 35:1 to 37:1, you would obtain a net profi of $37 for each $1 bet. If you bet $5, your net profit would be $185. (The casino makes its profit by paying only $175 instead of the $185 that would be paid with a roulette game that is fair instead of favoring the casino.)
P(A or B) = P(A) + P(B) - P(A and B), To find P(A or B), find the sum of the number of ways event A can occur and the number of ways event B can occur, adding in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space.
Events A and B are disjoint (or mutally exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.)