Question

Step 1: Players will spin a spinner with 5 different coloured sections. If the player spins RED they end the game with no prize. If they spin any other colour they continue to step 2.

Step 2: Players will roll two dice. If the sum of the two dice is 7+ then the player moves to step 3. If the player rolls a sum less than 7 then the player is out but will get $1 as a prize.

Step 3: Players will roll a single die and remember the number. They will then roll the die a second time. If they roll a different number then what they rolled the first time they are out. If they roll the same number twice they win the jackpot of $25.

1st: Calculate the theoretical probability for each step of the game

2nd: Create a tree diagram for the entire game

3rd: Figure out the probability for the four branches of the tree (As in the four points the game ends at)

Answer 1

Answer to 1^st Question:
In STEP 1 –

It is given that there are 5 different coloured sections of the spinner. If the player spins RED, he / she ends the game with no prize else one continues to Step 2

Probability of spinning red = 1/5 = 0.2

Probability of not spinning red = 1 – (1/5) = 4/5 = 0.8

The Theoretical probability of continuing to Step 2 = 4/5 = 0.8

In STEP 2 –

One moves to Step 3 if the sum of the 2 dice is ≥ 7

Total number of outcomes in a single 2 – dice throw = 6 x 6 = 36 (assuming both the dice to be 6 – sided fair dice)

The following table shows the possible sum when two dice are thrown -

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Therefore, Probability of getting a sum ≥ 7 = 21/36 = 7/12 = 0.5833

Probability of getting a sum < 7 = 1 – (21/36) = 15/36 = 5/12 = 0.4167

The Theoretical probability of continuing to Step 3 = 7/12 = 0.5833

In STEP 3 –

One will win the Jackpot if the same number is rolled twice.

Assuming the die to be fair,

Probability of rolling a similar number twice in two throws = 6/36 = 1/6 = 0.1667

The outcomes can be - {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

Probability of not rolling a similar number twice = 5/6 = 0.8333

The Theoretical probability of winning the Jackpot = 1/6 = 0.1667

Answer to 2^nd Question:
The Tree Diagram is given below –

(where the upper branch represents the probability of winning and the lower branch represents the probability of losing)

Answer to 3^rd Question:
Therefore,

Probability of game ending at STEP 1 = 1/5 = 0.2

(Total prize won $0)

Probability of continuing to Step 2 = 4/5 = 0.8

Total Probability that the game will end at STEP 2 = (4/5) x (5/12) = 1/3 = 0.3333

(Total prize won $1)

Probability of continuing to Step 3 = 7/12

Total Probability of losing the Jackpot = (4/5) x (7/12) x (5/6) = 7/18 = 0.3889

(Total prize won $25)

(All the probabilities are rounded up to 4 decimal places)