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1 Alice has five coins in a bag: two coins are normal (i.e., fair with one face Heads and the other face Tails), two are double-headed (i.e., both sides are Heads), and the last one is double-tailed (i.e., both sides are Tails). She reaches into the bag and randomly pulls out a coin, with each coin being equally likely to be selected. Without looking at the coin she drew, she tosses it once.
What is the probability that the side of the coin that lands face-down is Heads?
2 The coin lands and shows Heads face-up. Given this information, what is the probability that the face-down side is also Heads?
3Alice discards the coin used in the first round. She knows that it showed Heads face-up but does not look at the other side of the coin. She reaches again into the bag and draws out a second coin from the 4 remaining coins, with each coin equally likely to be selected. Again, without looking at this second coin, she tosses it once. Given her knowledge of the result of the first coin toss (i.e., that it landed Heads face-up), what is the probability that this second coin toss results in Heads face-up?
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To solve these probability problems, we need to use the concept of conditional probability. Conditional probability is the probability of an event happening given that we have some additional information.
Let's solve each of the questions one by one.
1. The probability that the side of the coin that lands face-down is Heads can be calculated using conditional probability. Let's denote the event of getting a Heads as H and the event of getting the side landing face-down as F. We want to find P(H|F), which is the probability of getting Heads given that the coin landed face-down.
Since there are five coins in the bag, and each coin is equally likely to be selected, the probability of selecting any particular coin is 1/5.
There are two normal coins, one double-headed coin, and one double-tailed coin. Out of these four coins, only two coins have Heads as one of their faces.
Therefore, the probability of getting a Heads when the coin lands face-down is 2/4, or 1/2.
Answer: The probability that the side of the coin that lands face-down is Heads is 1/2.
2. Given that the coin lands and shows Heads face-up, we want to find the probability that the face-down side is also Heads. Let's denote the event of the face-down side being Heads as D.
We know from the previous question that the probability of the side landing face-down as Heads is 1/2. However, now we have additional information that the coin shows Heads face-up.
So, out of the two possible outcomes where the coin shows Heads face-up, only one outcome has the face-down side as Heads.
Therefore, the probability that the face-down side is also Heads is 1/2.
Answer: The probability that the face-down side is also Heads is 1/2.
3. For this question, we have a new scenario where Alice discards the first coin used in the first round.
We already know from the previous questions that the probability of the side landing face-down as Heads is 1/2, and the probability of the face-down side being Heads given the coin shows Heads face-up is also 1/2.
Now, Alice reaches into the bag again and selects a second coin from the remaining four coins. The probability of selecting any particular coin is 1/4.
Given the result of the first coin toss (Heads face-up), this information does not provide any additional information about the second coin selected. The second coin toss is independent of the first coin toss.
Therefore, the probability that the second coin toss results in Heads face-up is still 1/2.
Answer: The probability that the second coin toss results in Heads face-up is 1/2.
I hope this helps you understand how to approach and solve probability problems using conditional probability.
1.) 3/5
2.) 2/3
3.) 13/24