Answer:
There is a 10.5% chance of having a positive payoff.
Step-by-step explanation:
The odds are 4 to 1 against, so we can estimate the probability of success (p) as
[tex]\frac{p}{q}=\frac{p}{1-p}=\frac{1}{4}\\\\4p=1-p\\\\5p=1\\\\p=0.2[/tex]
The expected pay for every success is 3 to 1, so we lose $1 for every lose and we gain $3 for every win.
The number of winnings in the 100 rounds to be even can be calculated as:
[tex]W+L=100\\\\L=100-W\\\\\\Payoff=0=3*W-1*L=3W-1*(100-W)=3W+W-100\\\\0=4W-100\\\\W=25[/tex]
We have to win at least 25 rounds to have a positive payoff.
As the number of rounds is big, we will approximate the binomial distribution to a normal distribution with parameters:
[tex]\mu=np=100*0.2=20\\\\\ \sigma=\sqrt{npq}=\sqrt{100*0.2*0.8}=4[/tex]
The z-value for x=25 is
[tex]z=\frac{X-\mu}{\sigma}=\frac{25-20}{4}=1.25[/tex]
The probability of z>1.25 is
[tex]P(X>25)=P(z>1.25)=0.10565[/tex]
There is a 10.5% chance of having a positive payoff.
NOTE: if we do all the calculations for the binomial distribution, the chances of having a net payoff are 13.1%.
