Probability of Joint Occurrences :
Another way to compute the probability of all three flipped coins landing heads is as a series of three different events: first flip the penny, then flip the nickel, then flip the dime. Will the probability of landing three heads still be .125?
Multiplication rule :
To compute the probability of two or more independent events all occurring— joint occurrence—multiply their probabilities.
For example, the probability of the penny landing heads is 1/2, or .5; the probability of the nickel next landing heads is 1/2, or .5; and the probability of the dime landing heads is 1/2, or .5; thus, note that
.5 x .5 x .5 = .125
which is what you determined with the classic theory by assessing the ratio of number of favorable outcomes to number of total outcomes. The notation for joint occurrence is P(AB) = P(A) × P(B) and reads: The probability of A and B both happening is equal to the probability of A times the probability of B.
Using the multiplication rule, you can also determine the probability of drawing two aces in a row from a deck of cards. The only way to draw two aces in a row from a deck of cards is for both draws to be favorable. For the first draw, the probability of a favorable outcome is 4/52. But because the first draw is favorable, only 3 aces are left among 51 cards. So the probability of a favorable outcome on the second draw is 3/51. For both events to happen, you simply multiply those two probabilities together:
Note that these probabilities are not independent. If, however, you had decided to return the initial card drawn back to the deck before the second draw, then the probability of drawing an ace on each draw is 4/52, as these events are now independent. Drawing an ace twice in a row, with the odds being 4/52 both times, gives the following:
In either case, you use the multiplication rule because you are computing probability for favorable outcomes in all events.Hope the above information was helpful.
