Union of probabilities(coin tosses)

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In summary, the probability of getting a head on the first toss or a head on the second toss or a head on the third toss can be calculated using the formula Pr(A or B or C) = Pr(A) + Pr(B) + Pr(C) - Pr(A and B) - Pr(A and C) - Pr(B and C) + Pr(A and B and C), where A, B, and C represent the events of getting a head on the first, second, and third toss respectively. This formula is based on the principle of inclusion-exclusion and can be used for any number of sets, not just two.
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torquerotates
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Homework Statement


Toss a coin 3 times. What is the probability that we get a head on the first toss or a head on the second toss or a head on the third toss?


Homework Equations


Pr(AorB)=Pr(A)+Pr(B)-Pr(AandB)



The Attempt at a Solution


A=head on 1st toss
B=head on 2nd toss
C=head on 3rd toss

Pr(AorBorC)=Pr(A)+Pr(B)+Pr(C)-Pr(A&B&C)

=(1/2)+(1/2)+(1/2)-(1/8)=11/8

but 11/8>1 which is a contradiction. So something is wrong with my sol.
 
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  • #2
torquerotates said:

Homework Statement


Toss a coin 3 times. What is the probability that we get a head on the first toss or a head on the second toss or a head on the third toss?


Homework Equations


Pr(AorB)=Pr(A)+Pr(B)-Pr(AandB)

This formula is true



The Attempt at a Solution


A=head on 1st toss
B=head on 2nd toss
C=head on 3rd toss

Pr(AorBorC)=Pr(A)+Pr(B)+Pr(C)-Pr(A&B&C)

This one isn't. The formula above only works as written when you have precisely two sets. You can expand it to a more general setting by considering what it means though. P(A or B) = P(A) + P(B) - P(A and B)

What this is really doing is just a counting argument. The number of things in the two sets A and B is just the number of things in A plus the number of things in B minus the number of things in both A and B. We had to subtract the last part because everything that was in A and B was counted when we counted the number of things in A, and also counted when we counted the number of things in B.

So similarly, to find P(A or B or C), we need the number of things in A, plus the number of things in B, plus the number of things in C. But in this case we double counted everything that is in A and B, and we double counted everything in B and C, and we double counted everything in A and C for the same reason as above. Also, anything in all three of A and B and C was counted three times, but when we remove everything in A and B, and everything in A and C, and everything in B and C, we uncounted those things three times also. So we need to add back in the number of things in all of A, B and C at the same time.

Do you see how to transform that into a formula about probabilities?
 

Related to Union of probabilities(coin tosses)

1. What is the Union of Probabilities in coin tosses?

The Union of Probabilities in coin tosses refers to the probability of a specific outcome occurring in a series of coin tosses. It is the combined probability of all possible outcomes.

2. How is the Union of Probabilities calculated in coin tosses?

The Union of Probabilities in coin tosses is calculated by adding the individual probabilities of each outcome. For example, if you are interested in the probability of getting at least one head in two coin tosses, you would add the probability of getting one head (1/2) to the probability of getting two heads (1/4), resulting in a Union of Probabilities of 3/4.

3. Can the Union of Probabilities in coin tosses be greater than 1?

No, the Union of Probabilities in coin tosses cannot be greater than 1. The maximum probability of an event occurring is 1, which represents a certain outcome. The Union of Probabilities is the sum of probabilities of all possible outcomes and cannot exceed the maximum probability.

4. How does the Union of Probabilities change with more coin tosses?

The Union of Probabilities in coin tosses increases with more coin tosses as the number of possible outcomes also increases. For example, the Union of Probabilities for getting at least one head in three coin tosses would be greater than the Union of Probabilities for getting at least one head in two coin tosses.

5. How can the Union of Probabilities be useful in real-world applications?

The Union of Probabilities can be useful in predicting the likelihood of events occurring in real-world situations. It can help in decision-making processes and risk assessment by providing a more comprehensive understanding of the potential outcomes. For example, the Union of Probabilities can be used in finance to calculate the probability of making a profit or loss in an investment portfolio.

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