Question

Suppose there exists a mini-lotto game, where each ticket costs $2, and a winning ticket allows you to cash it in for $1,000 (which goes directly in the winning player's pocket). Suppose that each ticket has a 1/600 chance of winning, and that all tickets are independent of each other, which in particular means that multiple wins are possible without changing the individual probability of obtaining a winning ticket. Right now, Person A and Person B both have $9,000 in their pockets. Person A decides that she is going to buy 1,800 tickets and person B decides that he is going to buy 3,600 tickets. Answer the following: (NOTE THAT THERE ARE PARTS A, B, C, AND D) a) (2 pts) What is the probability that, after all tickets are bought, all tickets are checked for winners, and all winners have been cashed in, Person A has exactly $10,400 in her pocket?

Answer 1

To find the probability that Person A has exactly $10,400 after buying 1,800 mini-lotto tickets, we use the binomial probability formula with the number of successes set to 5 (the number of winning tickets needed).

Step-by-step explanation:

To calculate the probability that Person A has exactly $10,400 after buying tickets in the mini-lotto game and cashing in all winners, we need to use the binomial probability formula. The following variables are given in the question: the cost per ticket ($2), the prize per winning ticket ($1,000), the probability of winning per ticket (1/600), and the number of tickets Person A buys (1,800).

Person A starts with $9,000 and spends $3,600 on tickets ($2 per ticket times 1,800 tickets), leaving her with $5,400. To have $10,400, she needs to gain $5,000, which means she needs exactly 5 winning tickets ($5,000/$1,000 per ticket).

We apply the binomial formula:

P(X = k) = nCk * pk * (1-p)n-k

Where n is the number of trials (tickets bought), k is the number of successes (winning tickets), and p is the probability of a single success.

So, P(X = 5) = 1800C5 * (1/600)5 * (599/600)1795

Using a calculator to compute the above probability, we get the probability that Person A has exactly $10,400 after the event.