Explanation:
To create a probability distribution for a coin flipping game, we toss a coin at least 25 times and record the number of heads and the number of tails. Let's assume we have the following data:
Number of Heads: 12
Number of Tails: 13
To calculate the probability distribution, we divide the frequency of each outcome by the total number of trials (in this case, 25).
For the number of heads:
P(X = 0) = 0/25 = 0
P(X = 1) = 0/25 = 0
P(X = 2) = 0/25 = 0
...
P(X = 12) = 1/25 = 0.04
...
P(X = 25) = 0/25 = 0
For the number of tails:
P(X = 0) = 0/25 = 0
P(X = 1) = 0/25 = 0
...
P(X = 13) = 1/25 = 0.04
...
P(X = 25) = 0/25 = 0
To show that this distribution meets the properties for a probability distribution, we need to ensure that the probabilities sum up to 1. In this case:
Sum of P(X = x) for all x = 0 + 0 + ... + 0.04 + ... + 0 = 1
To graph the distribution, we can use a bar graph where the x-axis represents the number of heads or tails, and the y-axis represents the probability.
The random variables in this data are the number of heads and the number of tails. In each trial, the outcome can be either a head or a tail, and the random variable represents the number of times each outcome occurs.
The values in the data may or may not come out as expected, depending on the fairness of the coin. If the coin is fair, we would expect the number of heads and tails to be close to each other. However, if the coin is biased, the distribution may be skewed towards one side.
The expected value of the distribution represents the average number of heads or tails we would expect over a large number of trials. In this case, the expected value for both heads and tails would be 12.5 (25 trials multiplied by the probability of getting a head or a tail, which is 0.5).
To set up a binomial distribution, we would need the probability of success (getting a head) and the number of trials (25 flips). In this case, the probability of success is 0.5. To find the probability that a head appears 10 times or less, we can sum the probabilities of getting 0, 1, 2, ..., 10 heads:
P(X ≤ 10) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 10)
By substituting the calculated probabilities from the probability distribution, we can find the desired probability.