Exercise 16 Page 716

We want to find the probability of getting exactly 4 heads while flipping a coin 12 times. This is a case of a binomial experiment. We have 12 repeated independent trials and each trial has only two outcomes, success or failure. In our case, landing on heads is interpreted as a success. The probability of success is denoted as p. Probability of Success p= 1/2 When flipping a coin, there are only two possible outcomes. Therefore, the probability of failure q is equal to 1-p. In this case, q is the same as p. Probability of Failure q= 1/2Now, let's recall the probability of exactly k successes in n trials for a binomial experiment. P(k successes)= _nC_k p^k q^(n-k) Since we will flip the coin 12 times, we know that the number of independent trials is n= 12. We are interested in getting exactly 4 heads, so k= 4. We can substitute all of these values into the formula and evaluate the right-hand side.

P(k successes)= _nC_k p^k q^(n-k)

SubstituteValues

Substitute values

P( 4 heads)= _(12)C_4 ( 1/2 )^4 ( 1/2 )^(12- 4)

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Simplify

SubTerm

Subtract term

P(4 heads)= _(12)C_4 ( 1/2 )^4 ( 1/2 )^8

MultPow

a^m*a^n=a^(m+n)

P(4 heads)= _(12)C_4 ( 1/2 )^(4+8)

AddTerms

Add terms

P(4 heads)= _(12)C_4 ( 1/2 )^(12)

WriteDec

Write as a decimal

P(4 heads)= _(12)C_4 (0.5)^(12)

Next, let's recall the formula to calculate _nC_k. _nC_k =n!/k!( n- k)! ⇒ _(12)C_4= 12! 4!( 12- 4)! We can substitute this into the expression on the right-hand side of our equation.

P(4 heads)= _(12)C_4 (0.5)^(12)

Substitute

_(12)C_4= 12!/4!( 12- 4)!

P(4 heads)= 12!/4!( 12- 4)! * (0.5)^(12)

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Evaluate right-hand side

SubTerms

Subtract terms

P(4 heads)= 12!/4!* 8! * (0.5)^(12)

Write as a product

P(4 heads)= 12*11*10*9*8!/4*3*2*1*8! * (0.5)^(12)

CancelCommonFac

Cancel out common factors

P(4 heads)= 12*11*10*9/4*3*2*1 * (0.5)^(12)

Multiply

Multiply

P(4 heads)= 11 880/24 * (0.5)^(12)

CalcQuot

Calculate quotient

P(4 heads)= 495 * (0.5)^(12)

UseCalc

Use a calculator

P(4 heads)= 0.120849...

RoundDec

Round to 3 decimal place(s)

P(4 heads) ≈ 0.121

We found the probability of flipping a coin 12 times and getting exactly 4 heads is about 0.121 or 12.1 %.