A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.) (b) Find the probability of getting exactly two heads. (c) Find the probability of getting two or more heads.

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A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.)   (b) Find the probability of getting exactly two heads.  (c) Find the probability of getting two or more heads.

comentezq · Accepted Answer

Step 1  It is given that n=3.  Here, probability of getting head or tail is 0.5.  The probability mass function of binomial distribution is given below:  P(X=x)=nCxpx(1−p)n−x  Step 2  b. The probability of getting exactly two heads is obtained as follows:  P(X=2)=3C2(0.5)2(1−0.5)3−2  =3!2!(3−2)!(0.5)2(0.5)⟨∵nCr=n!r!(n−r)!⟩  =0.375  Step 3  Thus, the probability of getting exactly two heads using calculator is 0.375.  The probability of getting exactly two heads using binomial probability distribution table is 0.375.  Step 4  c. The probability of getting two or more heads is obtained as follows:  P(X≥2)=P(X=2)+P(X=3)  =3C2(0.5)2(1−0.5)3−2+3C3(0.5)3(1−0.5)3−3  =3!2!(3−2)!(0.5)2(0.5)+3!3!(3−3)!(0.5)3(1)  =0.375+0.125  =0.500  Step 5  Thus, the probability of getting two or more heads using calculator is 0.500.  The probability of getting two or more heads using binomial probability distribution table is 0.500.

xleb123 · Accepted Answer

Answer: 0.5Explanation:(a) Find the probability of getting exactly two heads:Using the binomial distribution formula, the probability can be calculated as:P(X=k)=(nk)·pk·(1−p)n−kwhere:- P(X=k) is the probability of getting exactly k heads,- n is the number of trials (flips),- k is the number of desired successes (heads),- p is the probability of success on a single trial (probability of getting a head), and- (nk) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials.For this problem, we have n=3, k=2, and since it&#039;s a fair coin, p=0.5. Substituting these values, we get:P(X=2)=(32)·0.52·(1−0.5)3−2Calculating the above expression, we find the probability of getting exactly two heads is approximately 0.375.(b) Find the probability of getting two or more heads:To find the probability of getting two or more heads, we need to calculate the probability of getting exactly two heads and the probability of getting exactly three heads, and then add them together.Using the binomial distribution formula, the probability of getting exactly three heads is:P(X=3)=(33)·0.53·(1−0.5)3−3Calculating the above expression, we find that the probability of getting exactly three heads is approximately 0.125.Adding the probabilities of getting exactly two heads and exactly three heads, we get:P(X≥2)=P(X=2)+P(X=3)Substituting the calculated probabilities, we have:P(X≥2)=0.375+0.125Calculating the above expression, we find that the probability of getting two or more heads is approximately 0.5.

Jazz Frenia · Accepted Answer

Step 1:Let&#039;s denote the probability of success as p and the number of trials as n.In this case, since we are flipping a fair quarter, the probability of getting a head on any given flip is 0.5, so p=0.5. The total number of flips is 3, so n=3.Step 2:Now, let&#039;s solve the given problems:(b) Find the probability of getting exactly two heads.Using the binomial distribution formula, the probability of getting exactly two heads can be calculated as:P(X=k)=(nk)·pk·(1−p)n−k where X represents the random variable (number of heads), k represents the specific value we are interested in (2 heads), n is the total number of trials (3 flips), p is the probability of success (0.5), and (nk) represents the binomial coefficient.Substituting the values into the formula:P(X=2)=(32)·0.52·(1−0.5)3−2Simplifying:P(X=2)=3·0.52·(0.5)1P(X=2)=3·0.25·0.5P(X=2)=0.375So, the probability of getting exactly two heads is 0.375.Step 3:(c) Find the probability of getting two or more heads.To find the probability of getting two or more heads, we need to calculate the sum of probabilities for getting exactly two heads and getting exactly three heads.P(X≥2)=P(X=2)+P(X=3)We already found that P(X=2)=0.375. To find P(X=3), we use the same formula:P(X=3)=(33)·0.53·(1−0.5)3−3Simplifying:P(X=3)=(33)·0.53·0.50P(X=3)=1·0.53·1P(X=3)=0.125Now, we can calculate the probability of getting two or more heads:P(X≥2)=0.375+0.125P(X≥2)=0.5Therefore, the probability of getting two or more heads is 0.5.

fudzisako · Accepted Answer

(b) Find the probability of getting exactly two heads.Using the formula for the binomial distribution, the probability of getting exactly two heads in three flips of a fair quarter is given by:P(X=2)=(nx)·px·(1−p)n−xwhere n is the number of trials, x is the number of successful outcomes (in this case, the number of heads), and p is the probability of success (getting a head).In this problem, n=3, x=2, and p=12 since the quarter is fair.P(X=2)=(32)·(12)2·(1−12)3−2Simplifying the expression:P(X=2)=3·(12)2·(12)1P(X=2)=3·14·12P(X=2)=38Therefore, the probability of getting exactly two heads is 38.(c) Find the probability of getting two or more heads.To find the probability of getting two or more heads, we need to calculate the probability of getting exactly two heads and the probability of getting exactly three heads, and then sum them up.P(X≥2)=P(X=2)+P(X=3)Using the same values as before:P(X≥2)=38+P(X=3)Since there is only one outcome where we get three heads in three flips, the probability of getting three heads is:P(X=3)=(33)·(12)3·(1−12)3−3Simplifying the expression:P(X=3)=(33)·(12)3·(12)0P(X=3)=1·(12)3·1P(X=3)=18Substituting the values back into the equation:P(X≥2)=38+18P(X≥2)=48P(X≥2)=12Therefore, the probability of getting two or more heads is 12.

A fair quarter is flipped three times. For each of the following probabilities,

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