The numbers racket is a well‑entrenched illegal gambling operation in most large cities. In one version, you choose one of the 1000 three‑digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three‑digit number is chosen at random and pays off $ 600 $600 . The mean payoff for the population of thousands of bets is μ = 60 μ=60 cents. Joe makes one bet every day for many years.

p(w) = p = 1/1000 = 0.001

q = 1-p = 0.999

reward for winning = 600$

payoff for attempt = 0.6$

a.

on an avg. jose loose 0.4$ per hand,

so equation will be to find out one win per x hands,

-0.4 x = -0.6x + 600

so x = 3000.

Hence 1 win out of 3000,

P(losing 0.4$) = \binom{3000}{1}(0.001)^1(0.999)^{2999}

P(losing 0.4$) = 0.1492

b.

Probability of winning 0.4$ per winning

equation,

0.4 x = -0.6x + 600

so ,

x = 600

so out of 600 game he have to win 1 game.

P(winning 0.4$) = \binom{600}{1}(0.001)^1(0.999)^{599}

P(winning 0.4$) = 0.3295

c.

Probability of winning 0.6$ per winning

equation,

0.6 x = -0.6x + 600

so ,

x = 500

so out of 500 game he have to win 1 game.

P(winning 0.6$) = \binom{500}{1}(0.001)^1(0.999)^{499}

P(winning 0.6$) = 0.3034

Attached is the same solutions incase the above isn't understandable.

Finally in frustration, Austin wrote to the San Antonio town council in October 1833 encouraging them to join with other towns to organize a local government independent of Coahuila. The town council turned Austin’s letter over to the Mexican government. Mexican officials considered Austin’s words an act of treason.

Explanation: