Answer:
a) Ann has a 1/3 chance of winning in the first round
b) The chance of Ann winning for the first time in the fourth round is 8/81
c) The probability that Ann's first win occurs after the fourth round is 16/81
Step-by-step explanation:
a) Each strategy is played with a probability of 1/3. Given any strategy, there’s a 1/3 chance that Bill will choose the strategy that allows Ann to win. Consequently, the probability of Ann securing a victory in the first round (or any round) is
1/3 * 1/3 + 1/3 * 1/3 + 1/3 * 1/3 = 1/9 + 1/9 + 1/9 = 1/3.
Thus, the likelihood of Ann winning the initial round is 1/3.
b) The chances of Ann winning a round stand at 1/3; therefore, her chances of not winning are 2/3. This must happen three times before her first victory. Thus, the probability that Ann's first win occurs in the fourth round is
(2/3)³ * 1/3 = 8/81.
c) The first victory happens after the fourth round if she remains unsuccessful in the first four rounds, translating to a possibility of (2/3)⁴ = 16/81.
The correct answer is that you start with no refills, paying $2 for the first drink, and when you receive a refill, you pay an additional $1, totaling $3.
The equivalence exists because tenths represent one-tenth of a whole. Having 20 tenths means multiplying 20 by 1/10, resulting in 20/10, which simplifies to 2 wholes. Conversely, expressing 2 as 20 tenths yields the same amount.
