Probability of getting three heads in five tosses of two coins?

In summary, the two statements are essentially the same and refer to the number of trials and expected outcomes in a binomial distribution. However, there is a difference between the probability of a specific outcome and the average outcome over a number of trials.
  • #1
s0ft
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This statement seems to me to be similar to something like : the number of trials is 10 and the expected number of heads is 3. What is the difference between these two, if there is any? And although the second seems to be a simple case of binomial distribution, I wonder how one would go about tackling the first situation.
 
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  • #2
hi s0ft! :smile:
s0ft said:
This statement seems to me to be similar to something like : the number of trials is 10 and the expected number of heads is 3. What is the difference between these two, if there is any?

i'm tempted to say that if you toss two coins, you'll never get three heads, no matter how many times you toss them! :rolleyes:

ignoring that interpretation, yes the two statements mean exactly the same :smile:
 
  • #3
s0ft said:
This statement seems to me to be similar to something like : the number of trials is 10 and the expected number of heads is 3. What is the difference between these two, if there is any? And although the second seems to be a simple case of binomial distribution, I wonder how one would go about tackling the first situation.

You seem to be confusing the probability of a given outcome (three heads in five tosses) with the average over a number of tosses. Unless the coin is biased, the average after ten tosses should be five.
 

1. What is the probability of getting three heads in five tosses of two coins?

The probability of getting three heads in five tosses of two coins is 10/32, or 0.3125. This can be calculated by using the formula for binomial probability: P(x) = (nCx) * p^x * q^(n-x), where n is the number of trials, x is the number of successes, p is the probability of success in one trial, and q is the probability of failure in one trial. In this case, n = 5, x = 3, p = 0.5, and q = 0.5.

2. How does the probability change if the coins are biased?

If the coins are biased, meaning they have a different probability of landing on heads than 0.5, then the probability of getting three heads in five tosses will also change. The formula for binomial probability still applies, but p will be replaced with the biased probability of heads. For example, if one of the coins has a probability of 0.6 of landing on heads, then the overall probability would be calculated as (5C3) * 0.6^3 * 0.4^2, which equals 0.3456 or 34.56%.

3. What is the likelihood of getting three heads in a row?

The likelihood of getting three heads in a row is 1/8, or 0.125. This can be calculated by using the formula for geometric probability: P(x) = (1-p)^(x-1) * p, where x is the number of trials and p is the probability of success in one trial. In this case, x = 3 and p = 0.5. This means that you have a 12.5% chance of getting three heads in a row if you toss the coins three times in a row.

4. Can the probability of getting three heads in five tosses be greater than 1?

No, the probability of getting three heads in five tosses cannot be greater than 1. This is because probability is a measure of the likelihood of an event occurring, and it cannot exceed 1 or 100%. If the calculated probability is greater than 1, it is likely that the formula was not applied correctly.

5. How can we use probability to predict future outcomes of coin tosses?

Probability can be used to make predictions about future outcomes of coin tosses, but it cannot guarantee the exact result. For example, if the probability of getting heads on a fair coin is 0.5, it does not mean that every other coin toss will result in heads. However, over a large number of trials, the actual results will likely be close to the predicted probability. This is because probability is based on the law of large numbers, which states that as the number of trials increases, the observed results will approach the expected probability.

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