You randomly draw letter tiles from a bag containing the letters from the word INDEPENDENT.

1. You randomly draw an N. Then, without replacing the first letter, you pick a D. Tell whether the events are independent or dependent. Then find the probability that both events occur.
---> I think it's a dependent event. But I'm having trouble solving for the probability.

2. You randomly pick a T. You replace it. Then you pick an E. Find the probability that both events occur.

I know this is the second time I'm posting this question, and I'm really sorry, but it's kind of urgent :(

Thanks again!

1. Draws without replacement are generally dependent. The outcome of the second draw is dependent on that of the first.

Give a try for the calculation of probability, and post what you have.

2. Draws with replacement are generally independent. So calculate the probability of each event and combine them ... how?

1.

In the first draw, there are 3 N's out of 11 letters. So you should be able to figure out the probability P1 (if necessary, see your previous post).
For the second draw, there are 2 D's in the 10 letters left (without replacement).
The probability is therefore P2.
The probability that BOTH of them will occur is P1.P2.
Can you work that out and post me the result?

1. You randomly draw an N. Then, without replacing the first letter, you pick a D.

To determine if these events are independent or dependent, you need to consider if the outcome of the first event affects the probability of the second event.

In this case, the events are dependent since the first letter drawn (N) is not replaced before picking the second letter (D). The probability of drawing an N is affected by the fact that an N has already been drawn.

To find the probability that both events occur, you multiply the probabilities of each event.

The probability of drawing an N is 2/11 (since there are 2 N's and a total of 11 letters), and the probability of drawing a D, given that an N has already been drawn, is 1/10 (since there is 1 D left and a total of 10 letters after the N is removed).

So, the probability of both events occurring is: (2/11) * (1/10) = 2/110 = 1/55.

2. You randomly pick a T. You replace it. Then you pick an E.

Since the letter T is replaced before picking the second letter, these events are independent. The probability of picking a T and picking an E are not influenced by each other.

The probability of picking a T is 1/11 (since there is 1 T and a total of 11 letters), and the probability of picking an E is also 1/11 (since there is 1 E and a total of 11 letters).

To find the probability that both events occur, you multiply the probabilities of each event.

So, the probability of both events occurring is: (1/11) * (1/11) = 1/121.

No problem at all! Let's break down each scenario and solve them one by one.

1. In the first scenario, you randomly draw an N from the bag. Without replacing it, you then pick a D. To determine whether these events are independent or dependent, we need to check if the outcome of the first event affects the probability of the second event.

Since you did not replace the first letter (N), the probability of drawing an N on the second event changes. Hence, the events are dependent.

To find the probability that both events occur, we multiply the probability of the first event by the conditional probability of the second event given that the first event has already occurred.

Let's calculate it step by step:

- Probability of drawing an N on the first event: There are 3 Ns in the word INDEPENDENT, and a total of 11 letters. So the probability is 3/11.
- Conditional probability of drawing a D given that an N has already been drawn: After removing one N from the bag, there are 2 Ns left and a total of 10 letters. So the probability is 2/10.

To find the probability of both events occurring, we multiply these probabilities: (3/11) * (2/10) = 6/110 = 3/55.

Therefore, the probability that you randomly draw an N, and then without replacing it, pick a D is 3/55.

2. In the second scenario, you randomly pick a T, replace it, and then pick an E. Since you replace the first letter (T) back into the bag, the probability of picking a T on the second event remains the same. The events are independent.

To find the probability that both events occur, we can simply multiply the probabilities of each event occurring individually.

- Probability of drawing a T: There is 1 T in the word INDEPENDENT, and a total of 11 letters. So the probability is 1/11.
- Probability of drawing an E: There are 2 Es in the word INDEPENDENT, and a total of 11 letters. So the probability is 2/11.

To find the probability of both events occurring, we multiply these probabilities: (1/11) * (2/11) = 2/121.

Therefore, the probability that you randomly draw a T, replace it, and then pick an E is 2/121.

I hope this helps! Let me know if you have any further questions.