Compound Probabilities
Compound probabilities are probabilities connected by "and" or "or".
P(A and B) is read the probability of A and B.
P(A or B) is read the probability of A or B
The formula for calculating P(A and B) depends on the relationship between events A and B. When the occurance of event A does not affect the occurance of event B we say that the events are independent. The multiplication rule is as follows:
P(A and B ) = P(A) x P(B)..........independent events.
For example: Suppose a fair coin is tossed and a fair die is rolled. Find the probability of getting a tail on the coin and a 5 on the die.
The Solution: Note that the word and was used. This means that the multiplication rule is to be used. Next we must decide if getting a 5 on the die depends on getting a tail on the coin. In other words, are the two events independent? Our common sense tell us that the two events are independent. So
P(Tail and five) = P(tail) x P(five) = 1/2 x 1/6 = 1/12.
If the two events A and B are not independent, that is if the occurance of event A affects the occurance of event B we say the events are dependent. The multiplication rule is as follows:
P(A and B) = P(A) x P(B| given that A has occured) .....B dependent on A
P(A and B) = P(B) x P(A| given that B has occured)......A dependent on B
The probabilities following the multiplication sign are conditional probabilities.
For example: A Jar contains 3 red balls; 4 yellow balls; and 5 green balls. If two balls are drawn in succession without replacement, find the probability of drawing a red ball on the first draw and a green ball on the second draw.
The solution: Note the word and was used. This means that the multiplication rule is to be used. Next we must decide if the two events are dependent. Since a red ball was removed on the first draw this affects the probability of drawing a green ball on the second draw since the sample space has been reduced from 12 ball to 11 balls. So
P(red and green) = P(red) x P(green| red ball on first draw)
P(red and green) = 3/12 x 5/11 = 15/132 = 0.1136
The Addition Rule:
The formula for calculating P(A or B) depends on the relationship between events A and B. If events A and B cannot occur at the same time( e. g. they have no common outcomes), they are said to be mutually exclusive events. The addition rule is as follows:
P(A or B) = P(A) + P (B) ....... mutually exclusive events.
For example: Suppose a fair coin is tossed and a fair die is rolled. Find the probability of getting a tail on the coin or a 5 on the die.
The Solution: Note that the word or was used. This means that the addition rule is to be used. Next we must decide if the two events can occur at the same time or if the two events have outcomes in common.. In other words are the two events mutually exclusive? It is clear that the two events do not have outcomes that are common. So
P(5 or T) = P(5) + P(T) = 1/6 + 1/2 = 4/6 = 2/3 = 0.666.
If events A and B are not mutually exclusive or if they can occur at the same time then the formula is:
P(A or B) = P(A) + P(B) - P(A and B)......not mutually exclusive events
For example, Suppose a card is drawn from a standard deck of 52 cards. Find the probability of drawing a Queen or a Heart.
The solution: Note the word or was used. This means that the addition rule is to be used. Next we must decide if the two events can occur at the same time. In other words are the events mutually exclusive? If one card is drawn from a deck of 52 cards can the card be both a Heart and a Queen? The answer is of course yes. So
P(Heart or Queen) = P(Heart) + P(Queen) - P( Heart and Queen).
Now, what is P(Heart)? Since there are 13 Hearts in a standard deck of 52 cards the answer is 13/52
Next, what is P(Queen)? Since there are 4 Queens in a standard deck of 52 cards the answer is 4/52.
Finally, what is P(Heart and Queen)?. Since there is only one Queen of Hearts in a standard deck of 52 cards, the answer is 1/52. Therefore:
P(Heart or Queen)= 13/52 + 4/52 - 1/52 = 16/52 = 0.3077