Final answer:
To find the probabilities of winning or losing the scratch off tickets and expected values, we use the given probabilities for each outcome and apply geometric probabilities for the number of tickets played. We adjust calculations according to the modified strategy when the budget increases to $10.
Step-by-step explanation:
To answer the student's question about scratch off tickets, we need to calculate probabilities and expected values:
a) Probability of Winning or Losing
You win $20 with probability 1/10, get a free ticket with probability 1/2, and lose with probability 2/5. To find the likelihood of winning $20, let's note that getting a free ticket is an opportunity to try again. So, the probability of winning is the sum of the probabilities of winning on the first ticket, plus winning after any number of free tickets.
b) Expected Winnings
The expected winnings are calculated by multiplying each outcome by its probability and summing these products.
c) Expected Number of Tickets Played
The expected number of tickets played can be found using geometric probability, as getting a free ticket can be considered a 'success' in continuing to play.
d) Probabilities with $10
With an extra $5, there are more possible outcomes to consider, including winning after one or two losses and losing twice. We must calculate each of these probabilities, considering the new conditions.
e) Expected Winnings with $10
The expected winnings can again be found by multiplying each monetary outcome by its probability and summing these products under the new strategy.
f) Expected Number of Tickets with $10
The expected number of tickets when using the $10 strategy can be found similarly to part c, adjusting for the possibility of losing twice.