Cam and Dean are playing a game with counters and dice. Cam has some counters, each labelled 1 on one side and 2 on the other. He flips the counters and scores the total of all the numbers showing. Dean has one die and scores the number showing after he rolls it. The winner is the player with the higher score. The game is a draw if both scores are the same. All counters and dice are fair, that is, all faces are equally likely to come up.

a) Cam flips 2 counters while Dean rolls one 6 faced die.
i) Calculate the probability that both score 3 in a game. (Correct probability)
ii) Explain why Cam is more likely to lose. (Convincing explanation)

b) Who is more likely to win if Cam flips three counters while Dean rolls one normal die? (Correct person with explanation)

c) Dean rolls an octahedral die with faces labelled 1 to 8. Determine the number of counters Cam must flip to make this a fair game, that is, a game in which Cam and Dean have same chance of winning. (Correct number of counters with explanation)

a) flipping 2 counters, Cam's outcomes:

11 , 12 , 21, and 22 , two ways to get a 3
---> prob(3) = 2/4 = 1/2

flipping a die, outcomes
1,2,3,4,5,6 , one way to get a 3 --> prob(3) = 1/6

prob(both get a 3) = (1/2)(1/6) = 1/12

CA DE ---- Cam's loss ---> Dean's win
11 3,4,5,6,----- 4ways
12 4,5,6--------3ways
21 4,5,6--------3ways
22 5,6 ---------2ways
There are 12 ways for Dean to win

CA DE --- Cam's wins
11 1
12 1,2
21 1,2
22 1,2,3 -------a total of 8 ways

CA DE --- it's a tie
11 2
21 3
12 3
22 4 ------ 4 ways

Prob(Dean wins) = Prob(Cam loses) = 12/24 = 1/2
prob(Cam wins) = 8/24 = 1/3
prob(tie) = 4/24 = 1/6

(note: 1/2 + 1/3 + 1/6 = 1 )

b) repeat experiment

You shouldn't be asking for help in the maths challenge. It is against the rules.

Cheater! I am doing the maths challenge too and this isn't fair to the other students.

a)

i) To calculate the probability that both Cam and Dean score 3, we need to consider all possible outcomes for Cam's counters and Dean's die.

Possible outcomes for Cam's counters:
- Counter 1 shows 1 and Counter 2 shows 2 (total score = 1 + 2 = 3)
- Counter 1 shows 2 and Counter 2 shows 1 (total score = 2 + 1 = 3)
- Counter 1 shows 2 and Counter 2 shows 2 (total score = 2 + 2 = 4)
- Counter 1 shows 1 and Counter 2 shows 1 (total score = 1 + 1 = 2)

There are 4 possible outcomes for Cam.

Possible outcomes for Dean's die:
- Rolling a 1, 2, 3, 4, 5, or 6 (each with equal probability)

Only rolling a 3 will result in a score of 3 for Dean.

Since the outcomes for Cam and Dean are independent, we can multiply their probabilities to find the overall probability of both scoring 3:

Probability for Cam = 2 favorable outcomes / 4 total outcomes = 1/2
Probability for Dean = 1 favorable outcome / 6 total outcomes = 1/6

Overall probability = (1/2) * (1/6) = 1/12

ii) Cam is more likely to lose because there are more possible outcomes that result in a score less than or equal to 3 for Cam compared to Dean. In this game, the highest possible score for Cam is 4 (when both counters show 2), while the highest possible score for Dean is 6 (when he rolls a 6). The probability of scoring a 4 for Cam is 1/4, while the probability of scoring a 6 for Dean is 1/6. Since 1/4 is less than 1/6, the chances of Cam scoring higher than Dean are lower, making Cam more likely to lose overall in this game.

b) To determine who is more likely to win when Cam flips three counters and Dean rolls one normal die, we need to compare the number of possible outcomes that result in a higher score for each player.

Possible outcomes for Cam's counters:
- The total score can range from 3 to 6, since each counter can show either 1 or 2. This gives a total of 4 possible outcomes for Cam.

Possible outcomes for Dean's die:
- The total score can range from 1 to 6, since the die has 6 faces. This gives a total of 6 possible outcomes for Dean.

Since there are more possible outcomes for Dean than for Cam, Dean is more likely to win in this scenario.

c) Dean rolls an octahedral die with faces labeled 1 to 8. To determine the number of counters Cam must flip to make the game fair, we need to find the number of possible outcomes for both Cam and Dean where the highest score for each player is the same.

Possible outcomes for Cam's counters:
- The total score can range from 1 to 8, since each counter can show either 1 or 2. This gives a total of 8 possible outcomes for Cam.

Possible outcomes for Dean's die:
- The total score can range from 1 to 8, since the die has 8 faces. This gives a total of 8 possible outcomes for Dean.

To make the game fair, the number of possible outcomes for Cam and Dean must be equal. In this case, Cam needs to flip 8 counters to match the 8 possible outcomes of Dean's die, resulting in a fair game.