Rework problem 29 from section 1.4 of your text, involving the flipping of a coin A coin is flipped. If a heads is flipped, then the coin is flipped 2 more times and the number of heads flipped is noted; otherwise (i.e., a tails is flipped on the initial flip) then the coin is flipped 3 more times and the result of each flip (i.e., heads or tails) is noted successively. How many possible outcomes are in the sample space of this experiment?

To determine the number of possible outcomes in the sample space of this experiment, we need to analyze the two scenarios based on the result of the first coin flip. 1. **First Flip: Heads (H)** - If the first flip is heads, the coin is flipped 2 more times. - Each of these flips can result in either heads (H) or tails (T). - The number of outcomes for the 2 additional flips is \(2^2 = 4\) (since each flip has 2 possible outcomes). - The possible outcomes in this case are: - HHH - HHT - HTH - HTT 2. **First Flip: Tails (T)** - If the first flip is tails, the coin is flipped 3 more times. - Each of these flips can also result in either heads (H) or tails (T). - The number of outcomes for the 3 additional flips is \(2^3 = 8\). - The possible outcomes in this case are: - TTT - TTH - THT - THH - HTT - HTH - HHT - HHH Now, we can combine the outcomes from both scenarios: - From the first scenario (first flip is H): 4 outcomes - From the second scenario (first flip is T): 8 outcomes To find the total number of possible outcomes in the sample space, we add the outcomes from both scenarios: \[ \text{Total Outcomes} = 4 + 8 = 12 \] Thus, the total number of possible outcomes in the sample space of this experiment is **12**.