A large cooler contains the following drinks: 11 lemonades, 6 Sprites, 14 Cokes, and 7 root beers. You randomly pick two cans, one at a time (without replacement). Compute the following probabilities.
What is the probability that you get either two root beers or two lemonades?
(d) What is the probability that you get one can of Coke and one can of Sprite?
(e) What is the probability that you get two drinks of the same type?
(d) The probability of getting one can of Coke and one can of Sprite can be calculated as follows:
First, we need to calculate the total number of ways we can choose two cans from the cooler, which is given by the combination formula:
Total number of ways to choose 2 cans = (number of cans in the cooler) choose (2) = (11+6+14+7) choose (2) = 38 choose 2 = 703.
Next, we calculate the number of ways we can choose one can of Coke and one can of Sprite:
Number of ways to choose 1 Coke and 1 Sprite = (number of Cokes) * (number of Sprites) = 14 * 6 = 84.
Therefore, the probability of getting one can of Coke and one can of Sprite is:
P(Coke and Sprite) = (Number of ways to choose 1 Coke and 1 Sprite) / (Total number of ways to choose 2 cans) = 84 / 703 ≈ 0.1196.
(e) The probability of getting two drinks of the same type can be calculated by summing up the probabilities of getting two root beers and getting two lemonades:
P(Two root beers) = (Number of ways to choose 2 root beers) / (Total number of ways to choose 2 cans) = (Number of ways to choose 2 root beers) / 703.
P(Two lemonades) = (Number of ways to choose 2 lemonades) / (Total number of ways to choose 2 cans) = (Number of ways to choose 2 lemonades) / 703.
The total probability is then:
P(Two drinks of the same type) = P(Two root beers) + P(Two lemonades)
Unfortunately, the problem has not provided the number of root beers and lemonades in the cooler, so we cannot calculate the exact probabilities for this case.
To find the probabilities, we need to know the total number of possible outcomes and the number of favorable outcomes in each case.
Let's start with part (a) - What is the probability that you get either two root beers or two lemonades?
To calculate this probability, we need to find the number of favorable outcomes and divide it by the total number of possible outcomes.
1. Total number of possible outcomes:
Since you are picking two cans without replacement, there are a total of (11+6+14+7) = 38 cans in the cooler to choose from initially. After picking one can, there are 37 cans left to choose from, and after picking the second can, there are 36 cans left. Therefore, the total number of possible outcomes is 38*37 = 1406.
2. Number of favorable outcomes:
To get two root beers, there are 7 root beers to choose from initially, and after picking one root beer, there are 6 root beers left. So, the number of favorable outcomes for two root beers is 7*6 = 42.
Similarly, to get two lemonades, there are 11 lemonades to choose from initially, and after picking one lemonade, there are 10 lemonades left. So, the number of favorable outcomes for two lemonades is 11*10 = 110.
Therefore, the number of favorable outcomes for either two root beers or two lemonades is 42 + 110 = 152.
Now, we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 152 / 1406 ≈ 0.1084 or 10.84%
Moving on to part (b) - What is the probability that you get one can of Coke and one can of Sprite?
Similarly, we need to find the number of favorable outcomes and divide it by the total number of possible outcomes.
1. Total number of possible outcomes:
We have already calculated the total number of possible outcomes, which is 1406.
2. Number of favorable outcomes:
To get one can of Coke and one can of Sprite, we need to choose one Coke from the 14 available initially and one Sprite from the 6 available initially. Therefore, the number of favorable outcomes is 14*6 = 84.
Now, we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 84 / 1406 ≈ 0.0598 or 5.98%
Finally, for part (c) - What is the probability that you get two drinks of the same type?
Again, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
1. Total number of possible outcomes:
We have already calculated the total number of possible outcomes, which is 1406.
2. Number of favorable outcomes:
To get two drinks of the same type, we can either choose two cans of lemonade, two cans of Sprite, two cans of Coke, or two cans of root beer. The number of favorable outcomes for each type is calculated as follows:
- Two lemonades: 11*10 = 110
- Two Sprites: 6*5 = 30
- Two Cokes: 14*13 = 182
- Two root beers: 7*6 = 42
So, the total number of favorable outcomes for getting two drinks of the same type is 110 + 30 + 182 + 42 = 364.
Now, we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 364 / 1406 ≈ 0.2590 or 25.90%
Therefore, the probability of getting either two root beers or two lemonades is approximately 10.84%, the probability of getting one can of Coke and one can of Sprite is approximately 5.98%, and the probability of getting two drinks of the same type is approximately 25.90%.
Either-or probabilities are found by adding the individual probabilities.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
P(RB) = 7/38 * (7-1)/(38-1) = ?
P(L) = 11/38 * 10/37 = ?
P(2rb or 2L) = P(RB) + P(L)
(D) P(CS) = 14/38 * 6/(38-1) = ?
(E) P(2same) = P(RB) + P(L) + etc.