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Problem 10

# List the elements in the given set. The set of outcomes (a) of tossing three distinguishable coins and (b) of tossing three indistinguishable coins [HINT: See Example 1.]

Expert verified
(a) For three distinguishable coins, there are 8 possible outcomes: HHH, HHT, HTH, THH, HTT, TTH, THT, TTT. (b) For three indistinguishable coins, there are 4 possible outcomes based on the number of Heads and Tails: 3H 0T (HHH), 2H 1T (e.g., HHT), 1H 2T (e.g., HTT), 0H 3T (TTT).
See the step by step solution

## Step 1: Find the outcomes when tossing three distinguishable coins

These are three distinguishable coins, and each coin can have two outcomes: Heads (H) or Tails (T). Thus there are 2^3 = 8 possible outcomes. We can list all possible combinations as follows: 1. HHH (Coin 1: H, Coin 2: H, Coin 3: H) 2. HHT (Coin 1: H, Coin 2: H, Coin 3: T) 3. HTH (Coin 1: H, Coin 2: T, Coin 3: H) 4. THH (Coin 1: T, Coin 2: H, Coin 3: H) 5. HTT (Coin 1: H, Coin 2: T, Coin 3: T) 6. TTH (Coin 1: T, Coin 2: T, Coin 3: H) 7. THT (Coin 1: T, Coin 2: H, Coin 3: T) 8. TTT (Coin 1: T, Coin 2: T, Coin 3: T) So, the total number of outcomes for (a) is 8.

## Step 2: Find the outcomes when tossing three indistinguishable coins

In this case, we are only interested in the number of Heads and Tails, not the specific order of the coins. We can list all possible outcomes (counting only distinct outcomes) as follows: 1. 3H, 0T: Three Heads and zero Tails (Only one possibility: HHH) 2. 2H, 1T: Two Heads and one Tail (Three possibilities: HHT, HTH, THH) 3. 1H, 2T: One Head and two Tails (Three possibilities: HTT, THT, TTH) 4. 0H, 3T: Zero Heads and three Tails (Only one possibility: TTT) So, the total number of outcomes for (b) is 4.

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