Final answer:
The probability of having exactly one digit in a four-character password is calculated by considering position permutations and the available choices for letters and numbers. The expected value of the card drawing game is found by computing the probability-weighted average of all possible outcomes.
Step-by-step explanation:
The student has two separate questions regarding probability. The first part of the question asks for the probability that exactly one of four characters in a randomly generated password (consisting of lowercase letters and digits from 0 to 9) will be a number.
The second part asks for the expected value when drawing cards from a standard deck of 52 cards where drawing a face card earns points and other cards result in a loss. Calculating expected value involves multiplying each outcome by its probability and summing these products.
For the password situation, there are 26 lowercase letters and 10 digits, leading to 36 possible characters for each position in the password. The first character has a 10/36 chance of being a number, followed by a 26/35, 25/34, and 24/33 chance that the subsequent characters are letters. This process repeats for each of the four positions a number can be in, which then is summed up to find the final probability.
For the card game expected value calculation, the probability of drawing a face card is 12/52, and the probability of drawing a non-face card is 40/52. The expected value is calculated as follows:
- Winning outcome: 12/52 chance of winning 10 points
- Losing outcome: 40/52 chance of losing 2 points
The expected value (EV) is (12/52) * 10 + (40/52) * (-2).