Exercise 36 Page 745

Guessing on a multiple choice test is a binomial experiment. The success can be thought of as guessing the right answer to a question, and failure is guessing the wrong answer. The probability of success p is equal to 1 divided by number of questions — in this case, 3. Similarly, the probability of failure is q= 23.

Event	Probability
Success: Guessed Right Answer	p= 1/3
Failure: Guessed Wrong Answer	q= 2/3

The probability that we will guess at least 4 out of 5 questions right is equal to the sum of the probabilities that we will guess exactly 4 right answers and 5 right answers. P(At least 4right)=P(4right)+P(5right) Let's calculate the probabilities one at a time, starting with P(4right).

Probability of Four Correct Answers

To find the probability that we will guess 4 right answers on a 5-question long test, we can use the Binomial Probability Formula.

Binomial Probability
Suppose you have n repeated independent trials each with a probability of success p and a probability of failure q (with p+ q=1). Then the binomial probability of x successes in the n trials can be found by the following formula. P( x)= _nC_x p^x q^(n- x)

The test is 5 questions long, so n=5. We want to calculate the probability of choosing 4 correct answers, thus x= 4. The probabilities of success and failure are p= 13 and q= 23 respectively. Let's substitute! P( x)= _nC_x p^x q^(n- x) ⇕ P( 4)= _5C_4( 1/3)^4 ( 2/3)^(5- 4) The value of the combination _5C_4 can be found on Pascal's Triangle.

We can see that _5C_4 equals 5. Now we are ready to calculate the probability P(4right).

P(4right)= _5C_4 (1/3)^4 (2/3)^(5-4)

Substitute

_5C_4= 5

P(4right)= 5 (1/3)^4 (2/3)^(5-4)

▼

Evaluate right-hand side

SubTerm

Subtract term

P(4right)= 5 (1/3)^4 (2/3)^1

ExponentOne

a^1=a

P(4right)= 5 (1/3)^4 (2/3)

PowQuot

(a/b)^m=a^m/b^m

P(4right)= 5 (1^4/3^4) (2/3)

CalcPow

Calculate power

P(4right)= 5 (1/81) (2/3)

MultFrac

Multiply fractions

P(4right)= 5 (1*2/81*3)

Multiply

P(4right)= 5 (2/243)

MoveLeftFacToNum

a*b/c= a* b/c

P(4right)= (5*2/243)

Multiply

P(4right)= (10/243)

CalcQuot

Calculate quotient

P(4right)=0.0411522...

RoundDec

Round to 3 decimal place(s)

P(4right)≈ 0.041

WritePercent

Convert to percent

P(4right)≈ 4.1 %

The probability of guessing 4 correct answers is about 4.1 %.

Probability of Five Correct Answers

Now we want to calculate the probability of guessing 5 correct answers. Similar to the previous section, we will use the Binomial Probability Formula with n= 5, p= 13, and q= 23. However, in this case x= 5. Let's substitute! P( x)= _nC_x p^x q^(n- x) ⇕ P( 5)= _5C_5( 1/3)^5 ( 2/3)^(5- 5) To find _5C_5, we can again use Pascal's Triangle.