Graphical Analysis In Exercises 3–5, the histogram represents ...

Question

Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.

a. p = 0.25

b. p = 0.50

c. p = 0.75

Histogram displaying a binomial distribution with five trials, showing probabilities for values 0 to 5 on the x-axis.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says the histogram shown illustrates a binomial distribution with 6 trials. Identify the correct probability of success P from the options below. And below the text we're given our histogram that has our values for X plotted along the x-axis and our probabilities plotted along the Y axis. Now along our x-axis, we have several values of X. The first value is going to be X equal to 0, and it has a probability of 0.12. The next value of X is X equals 1, and it has a probability of 0.3. The next value is 2. That has a probability of 0.33. The next value for X is X equal to 3, and it has a probability of 0.18. The next value of X is X equal to 4, and it has a probability of 0.06. The next value for X is 5, and it has a probability of 0.01, and the file value for X is 6, and it has a probability of approximately 0. We're also given four possible choices as our answers. For choice A, we have 0.30. For choice B, we have 0.50. For choice C, we have 0.70, and for choice D, we have 0.80. Now, we're asked to identify the probability of success P given our histogram or our binomial distribution. Now, if we look at our histogram here, we see that the value for X that has the highest probability is when X is equal to 2. And since it has the highest probability, that means it appears most frequently, so that is going to be the mode of our distribution. And we can actually relate our mode to our probability success P for a binomial distribution. So recall that for a binomial distribution that our mode. is going to be equal to the floor of the quantity of N + 1 in quantity multiplied by P. Where in here is the number of trials, which in this case, N is going to be equal to 6. So, we can substitute in our values for the mode and for N. So we'll have 2 is equal to the floor. Of the quantity of 6 + 1 in quantity multiplied by P. That means that 2 is equal to the 4 of 7 multiplied by P. So, our 4 function here actually implies an inequality. That means that 2, Has to be less than or equal to 7%, which has to be less than 3. So to solve this for P, we'll just divide everything by 7, so we'll get 2 divided by 7 is less than or equal to P, which is less than or equal to 3 divided by 7. So, writing those fractions as decimals, we get that 0.29 is less than or equal to E, which is less than 0.43. So here we just rounded those uh decibels to two decimal places. And now we just need to look at our answer choices to see which of those values fall within this range for P. And that means that the only value that actually falls within this range is answer choice A, which is 0.30. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

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Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.
a. p = 0.25
b. p = 0.50
c. p = 0.75

Key Concepts

Binomial Distribution

Probability of Success (p)

Histogram Interpretation

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Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.a. p = 0.25b. p = 0.50c. p = 0.75