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At the county fair, there are many popular games to play. One of them is Flip to Spin or Roll. First, the player flips a coin. If a head comes up, the player gets to spin the big wheel, which has ten equal sections: three red, three blue, and four yellow. If the coin comes up tails, the player gets to roll a cube with three red sides, two yellow sides, and one blue side. If the player ends up with blue, that player wins a stuffed animal.

Suppose that you know that Tyler won a stuffed animal. What is the probability that he started off by flipping heads?



Answer :

soelolol837
Okay, let's think through this step-by-step:
1) The game has two possible outcomes for the coin flip: heads or tails.
2) If the coin flip is heads, the player gets to spin the wheel, which has 3 blue sections.
3) If the coin flip is tails, the player gets to roll a cube, which has 1 blue side.
4) The player wins a stuffed animal if they get blue.
5) We know that Tyler won a stuffed animal, so we want to find the probability that he started by flipping heads.
6) Let's call the event of flipping heads "H" and the event of winning a stuffed animal "W".
7) We want to find P(H|W), the probability of flipping heads given that the player won a stuffed animal.
8) Using Bayes' Theorem:
P(H|W) = (P(W|H) * P(H)) / P(W)
9) P(W|H) is the probability of winning given a heads flip, which is 3/10 (3 blue sections on the wheel).
10) P(H) is 1/2, since there are two equally likely outcomes for the coin flip.
11) P(W) is the total probability of winning, which is (3/10 * 1/2) + (1/6 * 1/2) = 5/12.
12) Plugging these values into Bayes' Theorem:
P(H|W) = (3/10 * 1/2) / (5/12) = 3/5.
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Answer:

There are two ways to win a stuffed animal:

1. Flip a coin and get heads, then spin the wheel and land on blue.

2. Flip a coin and get tails, then roll the cube and get blue.

Let's calculate the probability of each of these two scenarios:

Probability of flipping heads and landing on blue:

P(heads) = 1/2

P(blue) = 3/10

P(heads and blue) = P(heads) * P(blue) = (1/2) * (3/10) = 3/20

Probability of flipping tails and rolling blue:

P(tails) = 1/2

P(blue) = 1/6

P(tails and blue) = P(tails) * P(blue) = (1/2) * (1/6) = 1/12

The probability of winning a stuffed animal by flipping heads and landing on blue is 3/20, and the probability of winning a stuffed animal by flipping tails and rolling blue is 1/12. Therefore, the probability that Tyler won a stuffed animal by flipping heads is:

P(heads | blue) = P(heads and blue) / P(blue)

P(heads | blue) = 3/20 / (3/20 + 1/12)

P(heads | blue) = 3/20 / 11/60

P(heads | blue) = 3/20 * 60/11

P(heads | blue) = 9/11

Therefore, the probability that Tyler won a stuffed animal by flipping heads is 9/11