Question

A. A fair 6-face die will be rolled some number of times, and you win if it shows an ace (one spot) 20% or more of the time. Otherwise, you lose. Which is better for you to win: 60 rolls or 600 rolls? Explain

b. As in (a), but you win if the percentage of aces is 15% or more.
c. As in (a), but you win if the percentage of aces is between 15% and 20% inclusive.
d. As in (a), but you win if the number of aces is within 2 of 1/6 of the number of rolls. In other words, for 60 rolls, you win if you get 8 to 12 aces in 60 rolls. For 600 rolls, you win if you get 98 to 102 aces in 600 rolls.

Answer 1

In the game requiring a 20% or higher rate of aces, 60 rolls are better to win due to higher probability variations with fewer rolls, allowing for easier deviation from the theoretical probability.

Step-by-step explanation:

To determine which scenario is better for winning, we must understand the concept of probability and the law of large numbers. A fair six-sided die has a theoretical probability, P(rolling an ace) or P(A), of 1/6. To win the game, we need to roll aces at least 20% of the time. For 60 rolls, we expect 10% of the outcomes to be aces (6 rolls), while for 600 rolls, we expect about 100 aces (16.67%). However, due to probability variations, smaller numbers of rolls are more likely to deviate from the expected probability. So, 60 rolls would be better as it's easier to reach 20% aces with fewer rolls where probability can vary more wildly.

Answer 2

Answer: The complete question is found in the attachment

Step-by-step explanation:

Law of large numbers: The probability of occurrence of an event becomes closer to the theoretical probability as the number of trials increases

P(an ace) = 1/6

= 0.1667

= 16.67%

a) In 600 rolls, the value will be close to 16.67. compared to 60 rolls

Greater than 20% interval doesn't include 16.67%. So, for more than 20% ace, 60 rolls is better.

b) More than 15% interval includes 16.67. So, it is better to roll 600 times

c) The interval between 15% and 20% include 16.67% and hence, 600 rolls is better

d) Larger number of trials is better to get exactly 16

So, 600 rolls is better