Question

For each of the following, list the sample space and tell whether you think the events are equally likely: a) Roll two dice; record the sum of the numbers. b) A family has 3 children; record each child's sex in order of birth. c) Toss four coins; record the number of tails. d) Toss a coin 10 times; record the length of the longest run of heads.

Short answer

a) No, b) Yes, c) No, d) No.

Step-by-step solution

Understanding the Problem

We need to determine the sample space for each scenario and assess whether each outcome within that sample space is equally likely. Sample space represents all possible outcomes in a probability experiment.

Roll Two Dice - Sample Space

For two dice rolls, each die has 6 possible outcomes. The numbers that can be rolled range from 2 (1+1) to 12 (6+6). The sample space for the sums is: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Roll Two Dice - Equally Likely Assessment

The sum of 7 can be made in multiple ways (1+6, 2+5, etc.) compared to the sum of 2 or 12 (1+1, 6+6). Therefore, the events are not equally likely, as some sums have more combinations of dice rolls.

Family with 3 Children - Sample Space

Each child can be a boy (B) or a girl (G). Thus, the sample space consists of: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.

Family with 3 Children - Equally Likely Assessment

Assuming each gender is equally likely, the probability of each sequence is the same, so each event is equally likely.

Toss Four Coins - Sample Space

We count the number of tails that result: 0 tails, 1 tail, 2 tails, 3 tails, 4 tails. Thus the sample space is: 0, 1, 2, 3, 4.

Toss Four Coins - Equally Likely Assessment

The number of ways to get 0 tails (all heads) is 1, whereas the number of ways to get 2 tails is more (i.e., C(4,2)). Thus, the events are not equally likely.

Toss Coin 10 Times - Sample Space

The longest run of heads could be anywhere from 0 to 10, depending on the patterns of heads and tails in the sequence.

Toss Coin 10 Times - Equally Likely Assessment

Due to the variety of sequences, and potential for more sequences to result in smaller runs, not all runs are equally probable. Therefore, the events are not equally likely.

Key concepts

Equally Likely Events

An Equally Likely Event refers to outcomes that have the same probability of occurring.
Imagine you have a balanced coin and you are about to toss it. The coin is equally likely to land on heads or tails. This is because both outcomes have the same chance of occurring.
For an event to be equally likely, each outcome should have the same weight or chance of happening.

• When rolling a die, each of the six faces is equally likely to show up.
• For outcomes to be equally likely, the setup or experiment must be fair without any bias.

However, when we roll two dice and record their sum, not all sums occur with the same frequency. Some sums, like 7, have more combinations than others, like 2. This means not all outcomes are equally likely.
Understanding whether outcomes are equally likely is important when calculating probabilities.

Dice Probability

In rolling a pair of dice, each die has 6 faces numbered from 1 to 6. When two dice are rolled, the number of possible outcomes is 36 since each die is independent.
The sum of the numbers on the two dice can range from 2 (1+1) to 12 (6+6).

• The total combinations that result in a sum of 7 include: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). This makes it the most frequent sum with 6 combinations.
• On the other hand, sums like 2 (1,1) or 12 (6,6) have only one combination each and are thus less probable.

This demonstrates that sums in dice probability are not equally likely as some sums have multiple combinations while others have fewer. Understanding this variability helps in predicting dice outcomes and crafting strategies in dice games.

Coin Toss Outcomes

Tossing a coin might seem simple, but it's a classic example in probability. When you toss a coin, you can get heads (H) or tails (T), with both being equally likely in a fair coin. Things become interesting when you increase the number of coin tosses.

• If you toss four coins, each coin can land on heads or tails, resulting in possible outcomes like 0, 1, 2, 3, or 4 tails.
• The chance of each specific number of tails is determined by the combination formula "C(n, k)", which calculates how many ways k tails can occur in n tosses.

For instance, having exactly 2 tails can occur in different sequences (HTTT, HTHT, THTT, etc.), making it more likely than all heads or all tails, with just one sequence possibility each.
Thus, not all outcomes like specific number of tails are equally likely when tossing multiple coins.

Gender Probability in Families

Examining gender probability within a family can reveal interesting outcomes. Consider a family with three children. Each child can be a boy (B) or a girl (G), leading to a combination of various sequences.
Assuming that each child has an equal probability of being a boy or girl, the scenarios for a family of 3 children result in eight possible sequences: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.

• Each sequence is equally likely, as the probability for each child being a boy or girl is typically treated as 50-50.
• This symmetry makes outcomes regarding gender sequences equally likely.

However, if additional factors are introduced, such as genetic predispositions, the equal probability assumption might not hold. But under perfect conditions and theoretical assumption, the sequences for gender combinations remain equally probable.