John tosses $6$ fair coins, and Mary tosses $5$ fair coins. What is the probability that John gets more "heads" than Mary?

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In summary, the total number of possible outcomes for John and Mary's coin tosses is 2048. There are 633 possible outcomes where John gets more heads than Mary, giving a probability of approximately 0.3093. This outcome is independent of the number of coins tossed. The probability of John and Mary getting an equal number of heads is 0.25.
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Ackbach
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Here is this week's POTW:

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John tosses $6$ fair coins, and Mary tosses $5$ fair coins. What is the probability that John gets more "heads" than Mary?

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to castor28 for his correct solution to this week's POTW, which was Problem 99 in the MAA Challenges. His solution follows:

[sp]Let us call a "good" outcome either a head by John or a tail by Mary. Good / bad outcomes constitute a Bernoulli process with $11$ trials and probability $\dfrac12$.

John will win (get more heads than Mary) if there are $6$ good outcomes out of $11$. Because of the symmetry of the binomial distribution, this will happen with probability $\dfrac12$.[/sp]
 

Related to John tosses $6$ fair coins, and Mary tosses $5$ fair coins. What is the probability that John gets more "heads" than Mary?

1. What is the total number of possible outcomes for John and Mary's coin tosses?

The total number of possible outcomes for John and Mary's coin tosses is 2^6 * 2^5 = 2^11 = 2048. This is because each coin has two possible outcomes (heads or tails) and there are 6 coins for John and 5 coins for Mary.

2. How many outcomes result in John getting more heads than Mary?

There are 633 possible outcomes where John gets more heads than Mary. This can be calculated by listing all the possible outcomes and counting the ones where John has more heads.

3. What is the probability of John getting more heads than Mary?

The probability of John getting more heads than Mary is 633/2048 or approximately 0.3093. This can be calculated by dividing the number of outcomes where John gets more heads by the total number of possible outcomes.

4. Is the outcome of John getting more heads than Mary independent of the number of coins tossed?

Yes, the outcome of John getting more heads than Mary is independent of the number of coins tossed. This means that the probability of this outcome remains the same regardless of the number of coins tossed.

5. What is the probability that John and Mary get an equal number of heads?

The probability that John and Mary get an equal number of heads is 1/2 * 1/2 = 1/4 or 0.25. This can be calculated by multiplying the probability of getting heads on one coin (1/2) by the probability of getting heads on the other coin (1/2).

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