a Given that we roll a red number cube and a yellow one, we are asked to find the probability of getting 1 on the red cube and 1 on the yellow cube. Note that rolling one cube does not affect the result of the other one. Therefore, the events are independent. If A and B are independent events, P(AandB) is given as follows.
P(AandB)= P(A)* P(B)With this in mind, we first need to find the probability of P(red 1) and P(yellow 1).
P(red1)&= 1/6
P(yellow1)&= 1/6
According to the formula, we have now to multiply the above values to get P(red1and yellow1).
P(red1and yellow1)&= 1/6* 1/6
&⇓
P(red1and yellow1)&=1/36
b In a similar way, we can now find the probability of P(red 2and yellow2). These events are also independent, since the result of one cube does not affect the result of the other one. Additionally, P(red 2)= 16 and P(yellow2)= 16. Let's multiply these values to get P(red2and yellow2).
c Note that in Part A and Part B we found that the probability of both cubes getting 1 or 2 is 136. Following this pattern, we can see that the probability of getting the same number on both cubes is 136. Also, these events are mutually exclusive. Therefore, we can calculate the probability of rolling any matching pair of numbers by adding 136 six times.
P(any matching pair of numbers)
=
1/36+ 1/36+ 1/36+ 1/36+ 1/36+ 1/36
⇓
P(any matching pair of numbers)=1/6
