In exercises 1 and 2, use the following information. You randomly draw letter tiles from a bag containing the letters from the word PROBABILITIES.
1. You randomly select a B. You replace it. Then you choose a T. Tell whether the events are independent or dependent. Then find the probability that both events occur.
2. You randomly pick an I. Then without replacing the first I, you pick another I. Tell whether the events are independent or dependent. Then find the probability that both events occur.
I know that the first one is independent, and the second is dependent. But I don't get how to solve the second part. PLEASE help! Thanks..
in the first, the prob of getting the B is 2/13, replacing that and then picking a T has a prob of 1/13
so the prob of picking a B, followed by the T is
(2/13)(1/13) = 2/169
in the second you are not replacing the letter so for the second prob. there are only 12 letters left
so Prob. of an I, then the second I is (3/13)(2/12) = 1/26
To determine whether the events are independent or dependent, we need to consider whether the outcome of the first event affects the outcome of the second event.
In the first scenario, where you select a B and then replace it before selecting a T, the events are independent. This means that the outcome of selecting the B does not impact the probability of selecting a T. The reason for this is that each time you replace the letter, the original distribution of letters is restored, and the probability remains the same.
To find the probability that both events occur, you multiply the probabilities of each event happening individually.
The probability of selecting a B is 2/13 because there are two B's in the word PROBABILITIES and a total of 13 letters in the bag.
The probability of selecting a T is 1/13 because there is only one T left in the word after selecting a B.
To find the probability that both events occur, multiply these probabilities together: (2/13) * (1/13) = 2/169.
In the second scenario, where you select two I's without replacing the first I, the events are dependent. This is because once you select the first I, the probability of selecting a second I changes. The original distribution of letters is not restored, and the probability is affected.
To find the probability that both events occur, you multiply the probabilities of each event happening.
The probability of selecting the first I is 2/13 (because there are two I's in the word PROBABILITIES and a total of 13 letters in the bag).
After selecting the first I, there is only one I left in the word, making the probability of selecting a second I 1/12.
To find the probability that both events occur, multiply these probabilities together: (2/13) * (1/12) = 1/78.
So, in this case, the probability of both events occurring is 1/78.