Question

b) modify the game to give the other player a slight advantage. explain the new rules and show that the other player now has the advantage.
c) modify the game to make it fair to both players. explain and show that each player has an equal chance of winning.

1. what are the odds in favour of rolling each of the following totals with a standard pair of dice?

a) 7
b) doubles
c) less than 4
d) greater t 4

1. tomorrows forecast calls for a 40% chance of rain.

a) what are the odds in favour of a rainy day tomorrow?
b) what are the odds against a rainy day tomorrow?

Explanation:

Step1: Recall the formula for odds

The odds in - favour of an event $E$ is given by $\frac{P(E)}{1 - P(E)}$, and the odds against an event $E$ is given by $\frac{1 - P(E)}{P(E)}$, where $P(E)$ is the probability of the event $E$.

Step2: Calculate the odds in favour of a rainy day (question 7a)

Given that $P(\text{rain})=0.4$. Then the odds in favour of rain is $\frac{P(\text{rain})}{1 - P(\text{rain})}=\frac{0.4}{1 - 0.4}=\frac{0.4}{0.6}=\frac{2}{3}$.

Step3: Calculate the odds against a rainy day (question 7b)

The odds against rain is $\frac{1 - P(\text{rain})}{P(\text{rain})}=\frac{1 - 0.4}{0.4}=\frac{0.6}{0.4}=\frac{3}{2}$.

Answer:

7a. $\frac{2}{3}$
7b. $\frac{3}{2}$