Assume that the coin is flipped 10 times.
What is the probability that both heads and tails occur?
3/4
1/2
To calculate the probability of both heads and tails occurring when a coin is flipped 10 times, we need to determine the total number of possible outcomes and the number of favorable outcomes.
1. Total number of possible outcomes:
When a coin is flipped, there are two possible outcomes - either a head (H) or a tail (T). Since it is being flipped 10 times, each flip has 2 possible outcomes. Therefore, the total number of possible outcomes is 2^10 = 1024.
2. Number of favorable outcomes:
To determine the number of favorable outcomes, we need to consider that both heads and tails should occur at least once in the ten flips.
One way to approach this is to calculate the number of outcomes where both heads and tails occur at least once, and subtract it from the total number of outcomes to get the number of outcomes where only either heads or tails occur.
- Number of outcomes where both heads and tails occur at least once:
In this case, we can think of fixing one outcome as heads and the other as tails, and then flipping the remaining 8 coins. So, for each of the 2^8 = 256 outcomes when the remaining 8 coins are flipped, both heads and tails have already occurred at least once. Therefore, there are 2^8 = 256 outcomes where both heads and tails occur at least once.
- Number of outcomes where only either heads or tails occur:
To find this, we subtract the number of outcomes where both heads and tails occur at least once from the total number of outcomes. So, the number of outcomes where only either heads or tails occur = total number of outcomes - number of outcomes where both heads and tails occur at least once = 1024 - 256 = 768 outcomes.
3. Calculate the probability:
The probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.
So, the probability that both heads and tails occur when a coin is flipped 10 times is:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 256 / 1024 = 1 / 4 = 0.25 = 25%