An ordinary (fair) die is a cube with the numbers
1
through
6
on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.
Compute the probability of each of the following events:
Event
A
: The sum is greater than
5
.
Event
B
: The sum is divisible by
5
.
Answer
die a-> 1 2 3 4 5 6
die b
1
2
3
4
5
6
36 boxes
label them for the sum of number above and to the left
eg
2 3 4 5 ....7
3 4 5 7 ....8
4 5 ........9
etc
Now for example 6 of the boxes contains a sum less than 5
that means 30/36 are greater than five
of course 30/36 = 5/6
To compute the probabilities of events A and B, we need to analyze the possible outcomes of rolling a die twice and adding the face values.
Step 1: Determine the sample space
The sample space, denoted by S, is the set of all possible outcomes of the random experiment. When rolling a die twice, each roll can have a result of 1, 2, 3, 4, 5, or 6. Therefore, the sample space is the set of all possible sums of two die rolls: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
Step 2: Determine the favorable outcomes for event A
Event A consists of all outcomes where the sum is greater than 5. The favorable outcomes for event A are {6, 7, 8, 9, 10, 11, 12}. There are 7 favorable outcomes for event A.
Step 3: Determine the probability of event A
The probability of an event, P(event), is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, the probability of event A is P(A) = Number of favorable outcomes / Total number of possible outcomes = 7/11.
Step 4: Determine the favorable outcomes for event B
Event B consists of all outcomes where the sum is divisible by 5. The favorable outcomes for event B are {5, 10}. There are 2 favorable outcomes for event B.
Step 5: Determine the probability of event B
Similarly, the probability of event B is P(B) = Number of favorable outcomes / Total number of possible outcomes = 2/11.
Therefore, the probabilities of events A and B are:
P(A) = 7/11
P(B) = 2/11
To compute the probability of each event, we need to first determine the total number of possible outcomes and the number of favorable outcomes for each event.
Total Number of Outcomes:
When a fair die is rolled twice, the total number of possible outcomes is the product of the number of outcomes for each roll. Since each roll has 6 possible outcomes (numbers 1 through 6), the total number of outcomes for both rolls is 6 x 6 = 36.
Number of Favorable Outcomes for Event A (Sum greater than 5):
We can find the favorable outcomes for Event A by considering all possible pairs of numbers that sum to a value greater than 5. These pairs are: (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6). There are 21 favorable outcomes for Event A.
Number of Favorable Outcomes for Event B (Sum divisible by 5):
To find the favorable outcomes for Event B, we need to consider all possible pairs of numbers that sum to a value divisible by 5. These pairs are: (1, 4), (2, 3), (3, 2), (4, 1), (2, 5), (3, 4), (4, 3), (5, 2), (3, 6), (4, 5), (5, 4), (6, 3). There are 12 favorable outcomes for Event B.
Calculating the Probability:
The probability of an event is equal to the number of favorable outcomes divided by the total number of outcomes.
Probability of Event A = Number of Favorable Outcomes for Event A / Total Number of Outcomes = 21 / 36 = 7 / 12
Probability of Event B = Number of Favorable Outcomes for Event B / Total Number of Outcomes = 12 / 36 = 1 / 3
Therefore, the probability of Event A (sum greater than 5) is 7/12 or approximately 0.583, and the probability of Event B (sum divisible by 5) is 1/3 or approximately 0.333.