## Probability Histograms for Experiments### Instructions:For each experiment, fill in the blank with the letter corresponding to the probability histogram that matches the described quantity.---### Experiment 1:A gambling game has a 30% chance of winning $2 and a 70% chance of losing $1. We play the game 100 times and track the net amount won or lost. ________### Experiment 2:We flip a coin 40 times and keep track of the number of heads. ________---### Histogram Descriptions:- **(a)** A bell-shaped histogram centered around 30, with values ranging from approximately 0 to 100. - **(b)** A narrow, centered histogram with a peak at 0, showing values from roughly -50 to 200. - **(c)** A symmetric, triangular histogram peaking at around 20, with values ranging from 0 to 40.- **(d)** A bell-shaped histogram peaking at approximately 20, with values from about 0 to 40.- **(e)** Similar to (d), but with higher frequency, the peak is slightly more pronounced.- **(f)** A broad histogram centered around 0, with smooth, wide distribution extending from -150 to 200.---Consider the nature of each experiment when selecting the histogram:- Experiment 1 involves a mix of wins and losses, which typically results in a histogram with a possible range of positive and negative outcomes.- Experiment 2, involving a simple probability situation like coin flips, often generates a symmetric distribution.

The probability histogram represents the probability on the vertical axis corresponding to its value on the horizontal axis. It can be seen that all the histograms provided are symmetric and it is known that for a symmetric distribution, the values of mean, median, and mode are equal. Thus, it can be said that the point on the horizontal axis corresponding to the peak point (where the probability is maximum) is the mean, median, or mode for that particular histogram.It is given in Experiment 1 that the gambling game is played 100 times such that it has a 30% likelihood of winning the amount of $2 and a 70% likelihood of losing the amount of $1. That is, Amount Probability 2 (win) 0.30 -1 (loss) 0.70 Thus, the expected net amount won or lost is as follows: Expected amount =2×0.30+-1×0.70=-0.1≈-0 Let X be a random variable that denotes the net amount won or lost. So, for this experiment, it can be said that the minimum value of X can be -100, as the loss amount is $1 and the number of times the game is played is 100. Similarly, the maximum value of X can be 200. Now, from the diagram, it can be observed that histogram (a), (b), and (f) lies within the range of -100 to 200. Also, histogram (b), and (f) has a mean value of 0. Thus, for experiment 1, the probability histogram is (a), (b), and (f).In Experiment 2, it is given that a coin is flipped 40 times and the number of heads is observed. Let X be the number of heads. As it is not given that the coin is biased, it can be considered that it is a fair coin. That is, it is an unbiased coin where the probability of head or tail is 12. So, it can be clearly said that X follows a binomial distribution with parameters n=40 and p=12, where n denotes the total number of trails done and p denotes the probability of success (getting head). Now, the minimum and maximum values of X can be 0 (no head) and 40 (all head), respectively, in the experiment. The probability distribution of X can be written as follows: PX=x=nxpx1-pn-x=40x12x1-1240-x=40x1240, x=0,1,…,40 The probabilities for different values of X are as follows: X P(X=x) 0 4001240=40!0!×40-0!×1240=9.09495e-13 1 3.63798e-11 2 7.09406e-10 ⋮ ⋮ 19 0.11940 20 0.12537 21 0.11940 ⋮ ⋮ 38 7.09406e-10 39 3.63798e-11 40 9.09495e-13 Thus, it can be said that the distribution is symmetric with a maximum probability of 0.12537 for X=20. Now, from the diagram, it can be observed that histogram (c), (d), and (e) lies within a range of 0 to 40. Also, the maximum probability in all these histograms does not exceed 0.12537. Thus, for experiment 2, the probability histogram is (c), (d), and (e).