Exercise 38 Page 773

We are given that the odds in favor of Event A are equal to the odds against Event A. Recall that the odds in favor of an event is the ratio between the number of favorable outcomes m and the number of unfavorable outcomes n. Odds in Favor = favorable outcomes/unfavorable outcomes = m/n On the other hand, the odds against an event are equal to the reciprocal of the odds in favor. Odds Against = unfavorable outcomes/favorable outcomes = n/m According to the given information, the odds in favor of Event A are equal to the odds against Event A. Odds in Favor = Odds Against ⇕ m/n = n/m Let's now simplify the obtained equation. Keep in mind that n and m are nonzero positive numbers.

m/n = n/m

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Simplify

MultEqn

LHS * nm=RHS* nm

m/n * nm = n/m* nm

MoveRightFacToNum

a/c* b = a* b/c

mnm/n = nnm/m

SplitIntoFactors

Split into factors

m* n* m/n = n* n* m/m

CancelCommonFac

Cancel out common factors

m* n* m/n = n* n* m/m

SimpQuot

Simplify quotient

m* m = n* n

ProdToPowTwoFac

a* a=a^2

m^2 = n^2

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(m^2) = sqrt(n^2)

SqrtPowToNumber

sqrt(a^2)=a

m=n

It follows from the given information that m is equal to n. We can now calculate the probability of Event A. The probability is the number of favorable outcomes m divided by the total number of outcomes — the sum of m and n. P(Event A) = m/m + n Since m is equal to n, we can substitute m for n in the expression for the probability.

P(Event A) = m/m + n

Substitute

n= m

P(Event A) = m/m + m

SumToProdTwoTerms

a+a=2a

P(Event A) = m/2m

ReduceFrac

a/b=.a /m./.b /m.

P(Event A) = 1/2