a coin is flipped fifteen times how many elements are in the equally- likely sample space? how many different events exist? what is the probability of at most a head occurring? what are the odds at least one head OCCURS

To determine the number of elements in the equally likely sample space, we need to consider the possible outcomes for each flip. Since a coin has two possible outcomes (heads or tails), flipping a coin 15 times will result in 2^15 (2 raised to the power of 15) equally likely outcomes in the sample space. Therefore, there are 32,768 elements in the equally likely sample space.

Next, let's consider the number of different events that exist. An event represents a specific outcome or a combination of outcomes from the sample space. In this case, there are several events we can consider. For example, the event of getting exactly 8 heads, or the event of getting more than 10 tails. The number of different events can vary depending on what specific outcomes or combinations you are considering.

To calculate the probability of at most a head occurring, we need to determine the number of outcomes that have zero heads, one head, or any number of heads up to and including fifteen. This can be done by summing the individual probabilities for each possible number of heads.

The probability of getting exactly zero heads (all tails) can be calculated as (1/2)^15 = 1/32,768.

The probability of getting exactly one head can be calculated as (15C1)(1/2)^1(1/2)^14 = 15/32,768. Here, 15C1 represents the number of ways to choose 1 head from 15 flips, which is equal to 15.

To calculate the probability of getting at most one head, we sum the probabilities of getting zero heads and one head:

P(at most one head) = P(zero heads) + P(one head) = 1/32,768 + 15/32,768 = 16/32,768 = 1/2,048.

Finally, to calculate the odds of at least one head occurring, we can subtract the probability of no heads from 1:

Odds(at least one head) = 1 - P(zero heads) = 1 - 1/32,768 = 32,767/32,768.

Please note that in probability, "odds" represents the ratio of favorable outcomes to unfavorable outcomes, while "probability" represents the ratio of favorable outcomes to the total number of possible outcomes.