Introduction to Probability Distributions

Show your work for each problem.

1.
For a valid probability distribution, all probabilities must sum to:
1. a)100
2. b)1
3. c)0
4. d)It depends on the number of outcomes
Answer: 1
The sum of all probabilities in a distribution must equal 1 (or 100%). This ensures that one of the outcomes will definitely occur.
2.
What is the probability of rolling any number on a fair 6-sided die?
1. a)
2. b)
3. c)
4. d)
Answer:
A fair die has 6 equally likely outcomes. Each face has probability of being rolled.
3.
If the probabilities in a distribution are , , and , what probability is missing?
Answer: 0.1
. Since probabilities must sum to 1, the missing probability is .
4.
Which type of distribution has all outcomes with equal probability?
1. a)Poisson distribution
2. b)Uniform distribution
3. c)Normal distribution
4. d)Binomial distribution
Answer: Uniform distribution
A uniform distribution is one where all outcomes have equal probability. A fair die is an example - each number has probability .
5.
How many total outcomes are possible when rolling two dice?
Answer: 36
Die 1 has 6 options, Die 2 has 6 options. Total outcomes = .
6.
What is the probability of rolling a sum of 7 with two dice? Write as a simplified fraction (e.g., 1/6).
Answer: 1/6
There are 6 ways to roll a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). With 36 total outcomes: .
7.
When rolling two dice, which sum is MOST likely?
1. a)12
2. b)7
3. c)2
4. d)6
Answer: 2
Sum 7 has the most combinations (6 ways), making it the most likely. Sum 2 and 12 each have only 1 way.
8.
A bag contains 3 red, 2 blue, and 5 green marbles. Create a probability distribution for drawing one marble.
Answer: P(red)=0.3, P(blue)=0.2, P(green)=0.5
1. How many marbles are in the bag total? 10
2. What is P(red)? Write as a decimal. 0.3
3. What is P(blue)? Write as a decimal. 0.2
4. What is P(green)? Write as a decimal. 0.5
9.
Calculate the expected value when rolling a fair die.
Answer: 3.5
1. What is the probability of each outcome? 1/6
2. Calculate: 1
3. Calculate: 2.5
4. Add the two parts to get 3.5
10.
A game costs 5 euros to play. You win 20 euros if you roll a 6, otherwise you win nothing. What is the expected profit per game?
1. a) euros
2. b) euros (about euros)
3. c) euros
4. d) euros (about euros)
Answer: euros (about euros)
euros. You lose on average!
11.
A factory produces items with a 4% defect rate. In a batch of 250 items, what is the expected number of defective items?
Answer: 10
1. Convert 4% to a decimal 0.04
2. How many items are in the batch? 250
3. Calculate: 10
12.
A coin is biased so that P(heads) = 0.6 and P(tails) = 0.4. If heads wins 3 euros and tails loses 2 euros, what is the expected value per flip?
Answer: 1
euro. On average, you win 1 euro per flip!
13.
A lottery ticket costs 2 euros. You have a 1% chance to win 100 euros, 5% chance to win 10 euros, and otherwise win nothing. Is this a fair game?
Answer: No, expected profit is -0.5 euros
1. What is P(win 100)? 0.01
2. What is P(win 10)? 0.05
3. What is P(win 0)? 0.94
4. Calculate expected winnings: 1.5
5. Subtract the ticket cost from expected winnings -0.5
14.
Which statement about expected value is FALSE?
1. a)Expected value can be negative
2. b)Expected value is always one of the possible outcomes
3. c)Expected value represents the long-run average
4. d)Expected value is a weighted average of outcomes
Answer: Expected value is always one of the possible outcomes
FALSE: Expected value is NOT always a possible outcome. For a die, , but you cannot roll 3.5! Expected value is a theoretical average, not necessarily an achievable result.
15.
An insurance company knows that 2% of customers file claims averaging 8000 euros. They charge each customer a 200 euro annual premium. With 5000 customers, what is their expected annual profit?
Answer: 200000
1. Calculate total premium income: 1000000
2. How many customers are expected to file claims? 100
3. Calculate expected payouts: 800000
4. Calculate expected profit: Income - Payouts 200000