Suppose any one round of a gambling game pays 3 to 1 and the odds are 4 to 1 against you.

if you play for 100 rounds and bet one dollar on each round, what is the chance you will come out ahead?

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

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:

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:

The z-value for x=25 is

The probability of z>1.25 is

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%.

126.66 m²

step-by-step explanation:

the pertinent area formula is a = πr², where r is the radius.

here, a = π(6.35 m)² = 40.32π, or approximately 126.66 m²