Questions: For each of the following, list the sample space and tell whether you think the outcomes are equally likely. a) Toss 2 coins; record the order of heads and tails. b) A family has 2 children; record the number of girls. c) Flip a coin until you get a tail or 4 consecutive heads. d) Roll three dice; record the largest number. a) Which of the following is the sample space for recording the order of heads and tails when tossing 2 coins? Let H represent getting a head and T getting a tail. A. HH, HT, TT B. H, T, HH, HT, TH, TT C. H, T D. HH, HT, TH, T

Solution Steps

To solve the given problems, we need to determine the sample space for each scenario and assess whether the outcomes are equally likely.

a) Tossing 2 coins involves considering all possible combinations of heads (H) and tails (T) in sequence. The sample space includes all permutations of H and T for two coins.

b) For a family with 2 children, we consider the possible combinations of boys (B) and girls (G). The sample space is based on the number of girls in each combination.

c) Flipping a coin until you get a tail or 4 consecutive heads involves considering sequences of flips that end with a tail or reach 4 heads in a row. The sample space includes all such sequences.

d) Rolling three dice and recording the largest number involves considering the possible outcomes for the largest number when three dice are rolled. The sample space includes the largest number from each possible roll.

For the multiple-choice question, we need to identify the correct sample space for tossing 2 coins and recording the order of heads and tails.

When tossing 2 coins, the possible outcomes are: \[ \text{Sample Space} = \{(H, H), (H, T), (T, H), (T, T)\} \] This results in 4 outcomes. Since each outcome has an equal probability of occurring, we conclude that the outcomes are equally likely. Thus, \( \text{equally likely} = \text{True} \).

For a family with 2 children, we can represent boys as \( B \) and girls as \( G \). The possible combinations for the number of girls are: \[ \text{Sample Space} = \{0, 1, 1, 2\} \] This indicates that there are 0, 1, or 2 girls in the family. The outcomes are not equally likely since the combinations \( (B, B) \), \( (B, G) \), and \( (G, B) \) do not have the same probability. Thus, \( \text{equally likely} = \text{False} \).

The sequences generated by flipping a coin until a tail appears or 4 consecutive heads are: \[ \text{Sample Space} = \{HHHH, HHHT, HHT, HT, T\} \] The outcomes are not equally likely due to the varying lengths and probabilities of the sequences. Thus, \( \text{equally likely} = \text{False} \).

When rolling three dice, the largest number can be any integer from 1 to 6. The sample space for the largest number is derived from all possible rolls: \[ \text{Sample Space} = \{1, 2, 3, 4, 5, 6\} \] The outcomes are not equally likely since the probability of rolling a higher number increases with the number of dice. Thus, \( \text{equally likely} = \text{False} \).

The correct sample space for recording the order of heads and tails when tossing 2 coins is: \[ \text{Options} = \{A: \{HH, HT, TT\}, B: \{H, T, HH, HT, TH, TT\}, C: \{H, T\}, D: \{HH, HT, TH, T\}\} \] The correct option is \( A \) since it accurately represents the outcomes of tossing 2 coins.

Final Answer