Siko is a contestant on a TV game show called Win a Million. Each time
she answers a multiple-choice question correctly, she wins money. If she picks a wrong answer, she is eliminated. If Siko does not know the right answer, she can use one of the following Helping Hands:
• Quiz the Crowd: She can poll the audience. The crowd has an experimental probability of being correct 85% of the time.
• Double Up: She can give two answers, instead of just one. If either is correct she stays in the game.
• Rule One Out: One of the incorrect answers is removed, leaving three choices.
Suppose Siko encounters three questions in a row to which she does not know the answers.
a) Assuming that she can use each Helping Hand only once during the game, and only once per question, what is the best estimated probability Siko has of staying
alive through the three questions? What assumptions did you make.
b) How many more times is Siko likely to stay in the game if she uses all three Helping Hands than if she simply guesses at random on all three questions?
a) To estimate the probability of Siko staying alive through the three questions, we can consider each Helping Hand she has.
First, let's assume Siko uses the Helping Hands optimally:
1. Quiz the Crowd: The crowd has an 85% chance of being correct. So, if Siko uses this Helping Hand, there is an 85% chance she will get the correct answer. Let's assume she uses this Helping Hand on the first question.
2. Double Up: With this Helping Hand, Siko gives two answers. If either answer is correct, she stays in the game. Since there are four options in a multiple-choice question, she has a 2/4 = 1/2 or 50% chance of randomly guessing the right answer. Using Double Up, we can consider Siko's chances of staying alive after using Quiz the Crowd on the first question.
3. Rule One Out: This Helping Hand removes one incorrect answer, leaving three choices. So, Siko has a 1/3 or approximately 33.33% chance of randomly guessing the correct answer if she uses Rule One Out.
To estimate the overall probability of Siko staying alive through the three questions, we multiply the individual probabilities together:
Probability of staying alive = Probability of using Quiz the Crowd (85%) * Probability of using Double Up (50%) * Probability of using Rule One Out (33.33%)
Probability of staying alive = 0.85 * 0.5 * 0.3333 ≈ 0.141625 (approximately 14.16%)
Therefore, the estimated probability of Siko staying alive through the three questions is approximately 14.16%.
Assumptions made:
- We assumed that Siko used each Helping Hand optimally, minimizing the risk of elimination.
- The experimental probability of the crowd being correct (85%) was assumed to remain the same throughout the game.
b) To determine how many more times Siko is likely to stay in the game if she uses all three Helping Hands compared to random guessing, we can calculate the difference in probabilities.
Let's assume that if Siko guesses randomly on each question, she has a 1/4 or 25% chance of getting the correct answer for each question.
Using random guessing, the probability of staying alive through the three questions would be: (0.25)^3 = 0.015625 (approximately 1.5625%)
To find the difference, we subtract the probability of random guessing from the probability of using all three Helping Hands:
Difference = Probability of using all three Helping Hands (14.16%) - Probability of random guessing (1.5625%)
Difference ≈ 0.1416 - 0.015625 ≈ 0.125975 (approximately 12.60%)
Therefore, Siko is likely to stay in the game approximately 12.60% more times if she uses all three Helping Hands compared to randomly guessing on all three questions.
a) To find the best estimated probability of Siko staying alive through the three questions, we need to consider the probabilities of different scenarios.
Assumptions:
1. Siko has no prior knowledge of the answers.
2. Each question has four possible answers, one of which is correct.
3. The experimental probability of the crowd being correct when polled is 85%.
4. Siko can use each Helping Hand only once during the game and only once per question.
Scenarios:
1. Siko guesses randomly on all three questions without using any Helping Hand.
- Probability of getting a question right by random guessing = 1/4 = 0.25
- Probability of getting a question wrong by random guessing = 3/4 = 0.75
- Probability of staying alive through all three questions by random guessing = 0.25 * 0.25 * 0.25 = 0.015625
2. Siko uses all three Helping Hands strategically.
- Scenario 2.1: Siko uses Quiz the Crowd on the first question.
- Probability of the audience being correct = 0.85
- Probability of getting the first question right using Quiz the Crowd = 0.85
- Probability of getting the first question wrong using Quiz the Crowd = 0.15
- Siko has two more questions left and two Helping Hands.
- Scenario 2.2: Siko uses Double Up on the second question.
- Probability of getting the second question right with one correct answer = 1 - (probability of getting it wrong with both answers)
- Probability of getting the second question wrong with both answers = (probability of getting the first question wrong using Quiz the Crowd) * (probability of getting the second question wrong randomly)
- Probability of getting the second question wrong randomly = 0.75
- Probability of getting the second question wrong with both answers = 0.15 * 0.75 = 0.1125
- Probability of getting the second question right with one correct answer = 1 - 0.1125 = 0.8875
- Siko has one more question left and one Helping Hand.
- Scenario 2.3: Siko uses Rule One Out on the last question.
- Probability of getting the last question right after one incorrect answer is removed = 1 - (probability of getting it wrong randomly after one incorrect answer is removed)
- Probability of getting the last question wrong randomly after one incorrect answer is removed = 0.75
- Probability of getting the last question right after one incorrect answer is removed = 1 - 0.75 = 0.25
The best estimated probability of Siko staying alive through the three questions using all three Helping Hands is the product of the probabilities from each scenario:
Best estimated probability = (0.85 * 0.8875 * 0.25) = 0.1879375
Therefore, the best estimated probability Siko has of staying alive through the three questions is approximately 0.1879 (18.79%).
b) To calculate how many more times Siko is likely to stay in the game if she uses all three Helping Hands compared to random guessing on all three questions, we need to compare the probabilities of staying alive in each case.
1. Random Guessing: Probability of staying alive through all three questions = 0.015625
2. Using all three Helping Hands: Best estimated probability of staying alive through all three questions = 0.1879375
The difference in probabilities is:
Difference = Best estimated probability - Probability of random guessing
Difference = 0.1879375 - 0.015625
Difference = 0.1723125
Therefore, Siko is likely to stay in the game approximately 0.1723 (17.23%) more times if she uses all three Helping Hands compared to random guessing on all three questions.
To calculate the best estimated probability of Siko staying alive through the three questions, we need to consider the probability of her staying alive on each question under different scenarios.
Here are the assumptions made for the calculations:
1. Siko does not know the answers to any of the three questions.
2. Siko uses each Helping Hand only once during the game.
3. Siko uses each Helping Hand only once per question.
4. The experimental probability of the Quiz the Crowd helping hand being correct is 85%.
5. Siko is equally likely to choose any of the multiple-choice answers if she guesses randomly.
Let's calculate the probability of staying alive for each question:
Question 1:
- If Siko guesses randomly without using any Helping Hands, her probability of staying alive is 1/4 (assuming 4 answer choices).
- If Siko uses the Quiz the Crowd helping hand, the probability of staying alive is 85% (0.85).
- If she uses the Double Up helping hand, the probability of staying alive is 1 - P(both answers are wrong), which is 1 - (3/4 * 3/4) = 1 - 9/16 = 7/16.
- If she uses the Rule One Out helping hand, the probability of staying alive is 1/3.
Question 2:
- If Siko guesses randomly without using any Helping Hands, her probability of staying alive is again 1/4.
- If she uses the Quiz the Crowd helping hand, the probability of staying alive is 0.85.
- If she uses the Double Up helping hand, the probability of staying alive is 7/16.
- If she uses the Rule One Out helping hand, the probability of staying alive is also 1/3.
Question 3:
- If Siko guesses randomly without using any Helping Hands, her probability of staying alive is still 1/4.
- If she uses the Quiz the Crowd helping hand, the probability of staying alive is once again 0.85.
- If she uses the Double Up helping hand, the probability of staying alive is 7/16.
- If she uses the Rule One Out helping hand, the probability of staying alive is 1/3.
Now, let's calculate the best estimated probability of Siko staying alive through the three questions.
If Siko guesses randomly without using any Helping Hands, the probability of staying alive on all three questions is (1/4) * (1/4) * (1/4) = 1/64.
If Siko uses all three Helping Hands optimally, the probability of staying alive on each question is:
- For each question, she can either use Quiz the Crowd (0.85), Double Up (7/16), or Rule One Out (1/3).
- Let's assume she uses the optimal Helping Hand for each question:
1. She uses Quiz the Crowd (0.85).
2. She uses Double Up (7/16).
3. She uses Rule One Out (1/3).
- The overall probability of staying alive through all three questions is (0.85) * (7/16) * (1/3) = 0.1484375.
Therefore, the best estimated probability Siko has of staying alive through the three questions is approximately 0.1484. This is much higher than the probability of 1/64 when guessing randomly without using any Helping Hands.
To answer part (b) of the question, we can calculate the difference in the number of times Siko is likely to stay in the game between using all three Helping Hands and guessing randomly on all three questions.
If Siko guesses randomly without using any Helping Hands, the probability of staying alive on each question is 1/4. The overall probability of staying alive through all three questions is (1/4) * (1/4) * (1/4) = 1/64.
If Siko uses all three Helping Hands optimally, the overall probability of staying alive through all three questions is approximately 0.1484 (as calculated in part (a)).
The difference in probabilities is 0.1484 - 1/64 = 0.1406, which means Siko is likely to stay in the game around 0.1406 more times if she uses all three Helping Hands compared to guessing randomly on all three questions.