How to Solve Probability Problems
You can solve many simple probability problems just by knowing two simple rules:
- The probability of any sample point can range from 0 to 1.
- The sum of probabilities of all sample points in a sample space is equal to 1.
The following sample problems show how to apply these rules to find (1) the probability of a sample point and (2) the probability of an event.
Probability of a Sample Point
The probability of a sample point is a measure of the likelihood that the sample point will occur.
Example 1
Suppose we conduct a simple
statistical experiment. We flip a coin one time. The coin flip can have
one of two equally-likely outcomes - heads or tails. Together, these outcomes represent the
sample space of our experiment. Individually, each outcome represents a sample
point in the sample space. What is the probability of each sample point?
Solution: The sum of probabilities of all the sample points must equal 1. And the probability of getting a head is equal to the probability of getting a tail. Therefore, the probability of each sample point (heads or tails) must be equal to 1/2.
Example 2
Let's repeat the experiment of Example 1, with a die instead of a coin. If we
toss a fair die, what is the probability of each sample point?
Solution: For this experiment, the sample space consists of six sample points: {1, 2, 3, 4, 5, 6}. Each sample point has equal probability. And the sum of probabilities of all the sample points must equal 1. Therefore, the probability of each sample point must be equal to 1/6.
Probability of an Event
The probability of an event is a measure of the likelihood that the event will occur. By convention, statisticians have agreed on the following rules.
- The probability of any event can range from 0 to 1.
- The probability of event A is the sum of the probabilities of all the sample points in event A.
- The probability of event A is denoted by P(A).
Thus, if event A were very unlikely to occur, then P(A) would be close to 0. And if event A were very likely to occur, then P(A) would be close to 1.
Example 1
Suppose we draw a card from a deck of playing cards. What is the probability
that we draw a spade?
Solution: The sample space of this experiment consists of 52 cards, and the probability of each sample point is 1/52. Since there are 13 spades in the deck, the probability of drawing a spade is
P(Spade) = (13)(1/52) = 1/4
Example 2
Suppose a coin is flipped 3 times. What is the probability of getting two tails
and one head?
Solution: For this experiment, the sample space consists of 8 sample points.
S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}
Each sample point is equally likely to occur, so the probability of getting any particular sample point is 1/8. The event "getting two tails and one head" consists of the following subset of the sample space.
A = {TTH, THT, HTT}
The probability of Event A is the sum of the probabilities of the sample points in A. Therefore,
P(A) = 1/8 + 1/8 + 1/8 = 3/8