2 dice are thrown.
a. find the odds in favor of the following events:
1. getting a sum of 7
2. getting a sum greater than 3
3. getting a sum that is an even number
b. find the odds against each of the events in part a
To find the odds in favor of an event, we need to determine the number of favorable outcomes and the number of total outcomes. In this case, two dice are thrown, so each die can land on any number from 1 to 6.
a.
1. Getting a sum of 7:
The favorable outcomes are when the two dice land on (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), or (4, 3), i.e., there are 6 favorable outcomes.
The total number of outcomes is the product of the number of outcomes for each die, which is 6 x 6 = 36.
Therefore, the odds in favor of getting a sum of 7 are 6:36, which can be simplified to 1:6.
2. Getting a sum greater than 3:
To find the favorable outcomes, we need to determine all the possible sums that are greater than 3. These sums can be obtained by adding together the numbers on each die.
The favorable outcomes are 4, 5, 6, 7, 8, 9, 10, 11, and 12. There are 9 favorable outcomes.
The total number of outcomes is still 36.
Thus, the odds in favor of getting a sum greater than 3 are 9:36, which can be simplified to 1:4.
3. Getting a sum that is an even number:
The favorable outcomes are when the sum is 2, 4, 6, 8, 10, or 12, which are all even numbers. There are 18 favorable outcomes.
The total number of outcomes is 36.
Therefore, the odds in favor of getting a sum that is an even number are 18:36, which can be simplified to 1:2.
b.
The odds against an event are found by subtracting the odds in favor from the total odds (which is the same as the total number of outcomes).
1. Odds against getting a sum of 7:
Total odds = 36
Odds in favor = 6
Odds against = Total odds - Odds in favor = 36 - 6 = 30.
2. Odds against getting a sum greater than 3:
Total odds = 36
Odds in favor = 9
Odds against = Total odds - Odds in favor = 36 - 9 = 27.
3. Odds against getting a sum that is an even number:
Total odds = 36
Odds in favor = 18
Odds against = Total odds - Odds in favor = 36 - 18 = 18.
Therefore, the odds against each of the events in part a are:
1. Getting a sum of 7: 30:6, which can be simplified to 5:1.
2. Getting a sum greater than 3: 27:9, which can be simplified to 3:1.
3. Getting a sum that is an even number: 18:18, which can be simplified to 1:1 or 1:1.