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Magazine Chapter 14: Problem 2
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
Expert verified
a) No, b) Yes, c) No, d) No.
Step by step solution
01
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.
02
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.
03
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.
04
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.
05
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.
06
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.
07
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.
08
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.
09
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
Understanding whether outcomes are equally likely is important when calculating probabilities.
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.
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 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.
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.
Thus, not all outcomes like specific number of tails are equally likely when tossing multiple coins.
- 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.
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.
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.
