Reference no: EM132829234

Binomial Probabilities

The Probability of an event can be expressed as a binomial probability if its outcomes can be broken down into two probabilities, p, which is a success and q, which is a failure. Where p and q are complementary

p + q = 1

Thus,

q = 1 - p

You need to rewrite the probabilities in the less than or equal to form to use the function in EXCEL. We will use Excel to find Binomial Probabilities. The probabilities do need to be in the less than or equal to form to use Excel. This is very important.

Here are some common Binomial Probabilities and how they would get re-written to calculate in the less than or equal to form, to use Excel.
• Pr (X = j) type in the following into Excel "= Binom.Dist( j, n, p, FALSE)"
• Pr (X ≤ j) type in the following into Excel "= Binom.Dist( j, n, p, TRUE)"
• Pr(X < j) = Pr (X ≤ j - 1) type in the following into Excel "= Binom.Dist( j-1, n, p, TRUE)"
• Pr(X > j) = 1 - Pr (X ≤ j) type in the following into Excel "= 1 - Binom.Dist( j, n, p, TRUE)"
• Pr(X ≥ j) = 1 - Pr (X ≤ j - 1) type in the following "= 1 - Binom.Dist( j - 1, n, p, TRUE)"
• To find the expected value of X, type in the following into Excel "= n*p"
• To find the standard deviation of X, type in the following into Excel "= sqrt(n*p*q)"

In general, when you use Excel Binom.dist, the excel function wants you to enter in the following information:
• number_s = number of successes (usually, this is called X)
• trails = number of trials (usually, this is call n)
• probability_s = probability of success (usually, this is called p)
• cumulative = FALSE for exact probabilities, and TRUE for "less than or equal to" (≤ ) probabilities

Suppose during week 1, we asked 10 random people whether or not they believe in Magic. Out of 10 people, 3 said yes, and 7 said no. Based on this sample, let us assume that 30% (which is 3/10 * 100%) of people in the population believe in Magic. In other words, let us assume that p = 0.30.
1) If we were to find another random sample of 15 people, what is the probability that exactly 5 people will believe in Magic?

• number_s = here we want to find the probability that X = 5 (exactly 5)
• trails = The number of trails, or n, is 15 since we are randomly sampling 15 people
• probability_s = this is the probability of success, p, which we assumed is 0.30
• cumulative = FALSE for exact probabilities Type in the following into Excel:

Convert the answer to a percentage using the "%" button (marked yellow below), then increase the number of decimals using the <-0.00 button (marked green below)

You will get 20.61%. Therefore, we can conclude that if 30% of people believe in Magic, there is a 20.61% chance that exactly 5 out of 15 randomly selected people believe in Magic.

2) If we were to find another random sample of 15 people, what is the probability that fewer than 5 people will believe in Magic?

• number_s = here we want to find the probability that X < 5, which is the same thing as X being less than or equal to 4, X ≤ 4 (fewer than 5 = less than or equal to 4)
• trails = The number of trails, or n, is 15 since we are randomly sampling 15 people
• probability_s = this is the probability of success, p, which we assumed is 0.30
• cumulative = TRUE for "less than or equal to" (≤ ) probabilities Type in the following into Excel:

Convert the answer to a percentage using the "%" button, then increase the number of decimals using the <-0.00 button (as shown in the image under #1)

You will get 51.55%. Therefore, we can conclude that if 30% of people believe in Magic, there is a 51.55% chance that fewer than 5 out of 15 randomly selected people believe in Magic.

3) If we were to find another random sample of 15 people, what is the probability that at least 3 people will believe in Magic?

• number_s = here we want to find the probability that X ≥ 3. To find this, we need to use the complement, which is X ≤ 2. In other words, Pr(X ≥ 3) = 1 - Pr (X ≤ 2) (at least 3 = 1 - less than or equal to 2)
• trails = The number of trails, or n, is 15 since we are randomly sampling 15 people
• probability_s = this is the probability of success, p, which we assumed is 0.30
• cumulative = TRUE for "less than or equal to" (≤ ) probabilities Type in the following into Excel:

Convert the answer to a percentage using the "%" button, then increase the number of decimals using the <-0.00 button (as shown in the image under #1)

You will get 87.32%. Therefore, we can conclude that if 30% of people believe in Magic, there is a 87.32% chance that at least 3 out of 15 randomly selected people believe in Magic.

4) If we were to find another random sample of 15 people, what is the probability that more than 6 people will believe in Magic?

• number_s = here we want to find the probability that X > 6. To find this, we need to use the complement, which is X ≤ 6. In other words, Pr(X > 6) = 1 - Pr (X ≤ 6) (more than 6 = 1 - less than or equal to 6)
• trails = The number of trails, or n, is 15 since we are randomly sampling 15 people
• probability_s = this is the probability of success, p, which we assumed is 0.30
• cumulative = TRUE for "less than or equal to" (≤ ) probabilities Type in the following into Excel:

Convert the answer to a percentage using the "%" button, then increase the number of decimals using the <-0.00 button (as shown in the image under #1)

You will get 13.11%. Therefore, we can conclude that if 30% of people believe in Magic, there is a 13.11% chance that more than 6 out of 15 randomly selected people believe in Magic.

Attachment:- Binomial Probabilities.rar