Question

worksheet
name:
date:
directions: decide whether the experiment is a binomial experiment. if it is, identify a success, specify the values of n, p, and q, and list the possible values of the random variable x. if it is not a binomial experiment, explain why.

1. cyanosis is the condition of having bluish skin due to insufficient oxygen in the blood. about 80% of babies born with cyanosis recover fully. a hospital caring for five babies born with cyanosis. the random variable represents the number of babies that recover fully.

directions: find the indicated probabilities. if convenient, use technology to find the probabilities. you must show all work including calculator commands.

1. you are taking a multiple - choice quiz that consists of 5 questions. each question has 4 possible answers, only one of which is correct. to complete the quiz, you randomly guess the answer to each question. find the probability of guessing…

a. exactly 3 answers correctly
b. at least 3 answers correctly

1. 59% of men consider themselves professional baseball fans. you randomly select 10 men and ask each if he considers himself a professional baseball fan. find the probability that the number who consider themselves baseball fans is…

a. exactly 8
b. at least 8
c. less than 3

Explanation:

Problem 2a: Probability of exactly 3 correct answers

Step 1: Identify binomial parameters

We have a binomial experiment with \( n = 5 \) (number of questions), \( p=\frac{1}{4}=0.25 \) (probability of correct guess), \( q = 1 - p=0.75 \) (probability of incorrect guess). The binomial probability formula is \( P(X = k)=\binom{n}{k}p^{k}q^{n - k} \), where \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \). For \( k = 3 \):
First, calculate \( \binom{5}{3}=\frac{5!}{3!(5 - 3)!}=\frac{5!}{3!2!}=\frac{5\times4\times3!}{3!\times2\times1}=10 \)

Step 2: Calculate the probability

Substitute \( n = 5 \), \( k = 3 \), \( p = 0.25 \), \( q = 0.75 \) into the formula:
\( P(X = 3)=\binom{5}{3}(0.25)^{3}(0.75)^{5 - 3}=10\times(0.25)^{3}\times(0.75)^{2} \)
\( (0.25)^{3}=0.015625 \), \( (0.75)^{2}=0.5625 \)
\( P(X = 3)=10\times0.015625\times0.5625 = 10\times0.0087890625=0.087890625 \)

Step 1: Define "at least 3"

"At least 3" means \( X\geq3 \), so \( X = 3,4,5 \). We need to calculate \( P(X = 3)+P(X = 4)+P(X = 5) \)

Step 2: Calculate \( P(X = 4) \)

Using the binomial formula, \( \binom{5}{4}=\frac{5!}{4!(5 - 4)!}=\frac{5!}{4!1!}=5 \)
\( P(X = 4)=\binom{5}{4}(0.25)^{4}(0.75)^{1}=5\times(0.25)^{4}\times0.75 \)
\( (0.25)^{4}=0.00390625 \), so \( P(X = 4)=5\times0.00390625\times0.75 = 5\times0.0029296875 = 0.0146484375 \)

Step 3: Calculate \( P(X = 5) \)

\( \binom{5}{5}=\frac{5!}{5!(5 - 5)!}=1 \)
\( P(X = 5)=\binom{5}{5}(0.25)^{5}(0.75)^{0}=1\times(0.25)^{5}\times1 \)
\( (0.25)^{5}=0.0009765625 \), so \( P(X = 5)=0.0009765625 \)

Step 4: Sum the probabilities

We already know \( P(X = 3)=0.087890625 \) from part (a).
\( P(X\geq3)=0.087890625 + 0.0146484375+0.0009765625=0.103515625 \)

Step 1: Identify binomial parameters

We have \( n = 10 \) (number of men), \( p = 0.59 \) (probability of being a baseball fan), \( q=1 - 0.59 = 0.41 \). For \( k = 8 \), use the binomial formula \( P(X = k)=\binom{n}{k}p^{k}q^{n - k} \)
First, calculate \( \binom{10}{8}=\frac{10!}{8!(10 - 8)!}=\frac{10!}{8!2!}=\frac{10\times9\times8!}{8!\times2\times1}=45 \)

Step 2: Calculate the probability

Substitute \( n = 10 \), \( k = 8 \), \( p = 0.59 \), \( q = 0.41 \) into the formula:
\( P(X = 8)=\binom{10}{8}(0.59)^{8}(0.41)^{10 - 8}=45\times(0.59)^{8}\times(0.41)^{2} \)
Calculate \( (0.59)^{8}\approx0.01299 \), \( (0.41)^{2}=0.1681 \)
\( P(X = 8)=45\times0.01299\times0.1681\approx45\times0.0021836\approx0.09826 \) (using more precise calculation: \( (0.59)^8 = 0.59\times0.59\times0.59\times0.59\times0.59\times0.59\times0.59\times0.59\approx0.0129906 \), \( (0.41)^2 = 0.1681 \), \( 45\times0.0129906\times0.1681\approx45\times0.002183\approx0.0982 \))

Answer:

\( 0.0879 \) (rounded to four decimal places)

Problem 2b: Probability of at least 3 correct answers