# Solving Discrete Distribution: P(X = 0 & 1)

• Somefantastik
In summary, in order to generate a random variable X, equally likely to be 0 or 1 using a biased coin, you must continuously flip the coin until you get one head and one tail, and then set X = 0 if the final flip is a head and X = 1 if the final flip is a tail. This method ensures that P{X = 0} = P{X = 1} = 1/2, as desired.
Somefantastik
I need help getting started on this.

Want to generate a random variable X, equally likely 0, 1, using biased coin (heads probability p).

1. Flip coin, result is labled $$0_{1}$$
2. Flip coin, result is labeled $$0_{2}$$
3. $$0_{1} = 0_{2}$$=> return to step 1
4. $$0_{2}$$= heads => X = 0, $$0_{2}$$ = tails => X = 1

Show X is equally likely to be 0 or 1.

A good place to start would be showing P{X = 0} = P{X = 1}. Can you show me how to find those two probabilities?

Last edited:
In order to get past step 3, you have to have had one head and one tail on the two flips. Since the probs of HT and TH are equal, step 4 will give you 2 equiprobable results.

Keeping in mind that this is a biased coin, can someone please show me how to explicitly find P{X = 0}? I understand that it is equal to 1/2 but I need to see how to get there.

Prob(HT)=Prob(TH)=p(1-p). Prob(get to step 4) is 2p(1-p), therefore prob(X=0)=prob(X=1)=p(1-p)/(2p(1-p))=1/2.

Is there a simpler way to do this where you continuously flip a coin until the last 2 results are different, that sets X = 0 if the final flip is a head, X = 1 if final flip is a tail?

Somefantastik said:
Is there a simpler way to do this where you continuously flip a coin until the last 2 results are different, that sets X = 0 if the final flip is a head, X = 1 if final flip is a tail?
No. A sequence of heads followed by one tail has a different probability than a sequence of tails followed by one head. You always need to start fresh as described in your original statement.

## 1. What is a discrete distribution?

A discrete distribution is a probability distribution that describes the likelihood of each possible outcome of a discrete random variable. Unlike continuous distributions, which can take on an infinite number of values, discrete distributions have a finite or countably infinite number of possible outcomes.

## 2. What is P(X = 0 & 1) in discrete distribution?

P(X = 0 & 1) refers to the probability that a discrete random variable X takes on the values of both 0 and 1. This can also be written as P(X = 0,1) or P(X = {0,1}). It is important to note that in probability, the "&" symbol represents the intersection of events, not the logical "and" operator.

## 3. How do you solve for P(X = 0 & 1)?

To solve for P(X = 0 & 1), you will need to know the probability distribution of the random variable X. This can be represented in a table, graph, or mathematical equation. Then, you can simply find the intersection of the events X = 0 and X = 1 to determine the probability of both events occurring simultaneously.

## 4. What is the difference between P(X = 0 & 1) and P(X = 0) + P(X = 1)?

P(X = 0 & 1) and P(X = 0) + P(X = 1) represent two different probabilities. P(X = 0 & 1) is the probability of both events X = 0 and X = 1 occurring simultaneously, while P(X = 0) + P(X = 1) is the sum of the probabilities of each event occurring individually. In some cases, these probabilities may be equal, but in other cases, they may differ.

## 5. How is discrete distribution used in real life?

Discrete distributions are used in many real-life situations, such as predicting the outcome of a coin toss, rolling a die, or counting the number of people with a certain characteristic in a population. They are also commonly used in statistical analysis and modeling to make predictions and decisions based on discrete data.

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