The study of probability began when people began studying games of chance such as flipping coins, rolling dice, drawing cards from a deck, or drawing marbles from an urn. Problems from games of chance still provide the best models on which to base a study of elementary probability, and we will concentrate on these problems.
It is customary to call activities such as flipping coins, rolling dice, blindly selecting cards from a deck, and drawing marbles from an urn experiments and to call the individual results outcomes.
We call the set of equally probable outcomes the sample space for the experiment. A toss of a fair coin has two equally probable outcomes. Thus, the sample space for a coin toss is heads or tails, as shown below.
H T
The roll of a single die has six equally probable outcomes. Thus, the figure below shows the sample space for the roll of a single die.
1 2 3 4 5 6
We define the probability of a particular even as the number of outcomes that satisfy the requirement divided by the total number of outcomes in the sample space. particular event=number outcomes that satisfy requirement
total number of outcomes in sample space
The probability of any event is a number between 0 and 1 inclusive. If no outcomes satisfy the requirement, the probability is 0, and if every outcome satisfies the requirement, the probability is 1.
Thus we see that a probability of -2 of 7 ½ is not possible because the probability of any event must be a number between 0 and 1.
Example:
A fair coin is tossed three times and comes up heads every time. What is the probability that on the next toss it will come up heads?
Example:
Six green marbles and eight red marbles are placed in an urn. One marble is drawn and then dropped back in the urn. Then a second marble is drawn and dropped back into the urn. Both marbles were red. If another marble is drawn, what is the probability that it will be red?
Practice:
A single die is rolled three times. The results are 1, 4, and 3, in that order. What is the probability that the next roll will produce a number greater than 2?
Practice:
Two dice are rolled. What is the probability that the sum of the numbers rolled is
a) 7
b) A number greater than 8
Designated Order:
The probability of future outcomes of independent events happening in a designated order is the product of the probability of the individual outcomes.
For example, if we toss a coin twice, the probability of getting a heads on the first toss and a tails on the second toss is one fourth.
P(H, T) = P(H) x P(T) = ½ x ½ = ¼
Example:
A fair coin is tossed four times. What is the probability that the first two times it comes up heads and the last two times it comes up tails?
Practice:
The spinner show is spun twice. What is the probability that the spinner stops on 4 and then on 3?