# Questions from a probabilty final

• SNOOTCHIEBOOCHEE
In summary, the conversation discussed two questions related to probability and random variables. The first question involved finding the probability that the number of flips required in the first trial is less than the number required in the second trial. The second question involved finding the density of a specific random variable. The conversation provided steps and formulas for solving both questions.
SNOOTCHIEBOOCHEE

## Homework Statement

Few questions here, nothing super tough, just can't get it/ want verification.

1. The following experiment is repeated twice: a fair coin is flipped repeatedly until it lands heads. Let X be the number of flips required in the first trial and Y the number required in the second trail. Find probability that X<Y.

2. Let U be a uniform (0,1) random variable. Find the density of ln(1/U)

## The Attempt at a Solution

1. So i figured the way to approach this problem is find P(X=Y) then we would calculate 1-P(X=Y)= P(X>Y or X<Y) then divide that by two to get P(X<Y).

So i wanted to calculate P(X=Y). i wasnt sure how to approach this, so i fixed X at some number k then found the probability that Y=k, which is 1/2^k. Is this right?

cause then the answer we get is 1-(1/2)^k/2. But I am not convinced of that answer.

2. This seems like it would be a one to one substitution.

Since U is uniform on (0,1) its density function is just 1.

Y=ln(1/u)
e^-Y= U

dy/du= -1/U

So our formula is 1/|-1/U|= U = e^-Y

This correct?

I can help you with your questions. For question 1, you are on the right track. To find P(X=Y), you can use the geometric distribution formula P(X=k) = (1/2)^k. So for P(X=Y), you would plug in k for both X and Y, which gives you (1/2)^k * (1/2)^k = (1/2)^(2k). This is the probability that both X and Y require the same number of flips. To find P(X<Y), you can use the same formula but plug in k+1 for Y, since X will always be less than Y. This gives you (1/2)^k * (1/2)^(k+1) = (1/2)^(2k+1). To find the final probability, you would subtract P(X=Y) from P(X<Y), which gives you (1/2)^(2k+1) - (1/2)^(2k) = (1/2)^(2k) * (1/2) - (1/2)^(2k) = (1/2)^(2k) * (1/2-1) = (1/2)^(2k) * (-1/2) = -(1/2)^(2k+1). This is the probability that X is less than Y, so to get the final answer, you would divide this by 2, giving you -(1/2)^(2k+2) = -(1/4)^k. This is the final probability that X is less than Y.

For question 2, your approach is correct. The density of U is 1, as you mentioned. To find the density of ln(1/U), you would use the formula you derived, which is 1/|(-1/U)| = U. This is the correct density function for ln(1/U).

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

## 1. What is the definition of probability?

Probability is the measure of the likelihood of a particular event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

## 2. How is probability calculated?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

## 3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual data collected from experiments or observations.

## 4. How do you calculate the probability of multiple events occurring?

To calculate the probability of multiple events occurring, you can use the multiplication rule. This states that the probability of two or more independent events occurring together is equal to the product of their individual probabilities.

## 5. Can probability be used to predict outcomes?

No, probability cannot be used to predict outcomes with certainty. It can only provide an estimate of the likelihood of an event occurring based on available information.

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