Question

For each of the following, list the sample space and tell whether you think the events are equally likely: a) Toss 2 coins; record the order of heads and tails. b) A family has 3 children; record the number of boys. c) Flip a coin until you get a head or 3 consecutive tails; record each flip. d) Roll two dice; record the larger number.

Short answer

a) Yes; b) No; c) No; d) No

Step-by-step solution

Determine Sample Space for Tossing 2 Coins

When tossing 2 coins, each coin can land on heads (H) or tails (T). The sample space consists of all possible outcomes based on the order of results: HH, HT, TH, TT. Thus, the sample space is \( \{HH, HT, TH, TT\} \). Since each coin toss is independent and has 2 possible outcomes, these events are equally likely.

Determine Sample Space for 3 Children - Number of Boys

Each child can be either a boy (B) or a girl (G). However, we're interested in the number of boys. The possible outcomes based on the number of boys are: 0 boys (GGG), 1 boy (BGG, GBG, GGB), 2 boys (BBG, BGB, GBB), 3 boys (BBB). Thus, the sample space is \( \{0, 1, 2, 3\} \) where the numbers represent the count of boys. These events are not equally likely because the number of combinations varies for 0, 1, 2, and 3 boys.

Determine Sample Space for Flipping Coin Until Head or 3 Tails

Flipping a coin until a head appears or 3 tails occur can have several outcomes: H, TH, TTH, TTT. The sample space includes these sequences: H (head on first flip), TH (head on second flip), TTH (head on third flip), and TTT (3 tails before any head). These events are not equally likely since the probability of achieving result in varying flips differ.

Determine Sample Space for Rolling 2 Dice - Larger Number

When rolling two dice, the larger number could be any number from 1 to 6. Each pair (a, b), where a and b are the numbers on the dice, will dictate the larger number. The sample space for the larger number is \( \{1, 2, 3, 4, 5, 6\} \), but these results are not equally likely. For instance, getting a larger number as 1 is less likely as compared to a 6, since a 1 can only occur if both dice show a 1, whereas a 6 occurs frequently unless both dice land less.

Key concepts

Sample Space

The concept of 'Sample Space' refers to the set of all possible outcomes of a random experiment. It's like an inventory list that includes every possible result you can get from a particular situation. For instance, think about tossing two coins. There are four possible outcomes: both could be heads, both could be tails, the first could be heads followed by tails, or tails followed by heads. If we write that in terms of a sample space, it would look like this: \( \{HH, HT, TH, TT\} \).

In probability, laying out all potential outcomes helps in calculating probabilities of individual events. Essentially, a sample space allows us to visualize all possible scenarios in advance, making it an effective tool for prediction and analysis.

Equally Likely Events

Equally likely events occur when each possible outcome of an experiment has the same chance of happening. When you toss a fair coin, getting heads or tails is equally likely because both outcomes have a 50% chance.

Consider an example of tossing two coins. The sample space is \( \{HH, HT, TH, TT\} \). Assuming the coins are fair, each of these outcomes is equally likely, making each probability equal to 25%. However, this concept doesn't apply universally. For instance, in a typical family with three children, the number of boys can be 0, 1, 2, or 3. These events aren't equally likely since the probability depends on detailed combinations, like specific birth orders.

Independent Events

Independent events in probability are scenarios where the outcome of one event does not affect the outcome of another. Each event happening is on its own and doesn't interfere with any other events. A classic example is rolling two dice. The result of the first roll doesn't affect the result of the second.

Think about flipping a coin multiple times. The outcome of each flip is independent of the others. This is why in the exercise, when we toss two coins, each toss is independent. The first coin doesn't influence the result of the second. This clarity helps in understanding why each outcome in the sample space remains equally likely.

Outcome Analysis

Outcome analysis involves studying possible results to understand probabilities more clearly. It often includes comparing different outcomes and evaluating their likelihood. This analysis helps in identifying whether events are equally likely or if some outcomes have a higher chance.

For example, consider rolling two dice and recording the larger number. The possible results are numbers from 1 to 6. However, these outcomes aren't equally likely. Getting a larger number of 1 is rare because it only occurs if both dice show 1. Outcomes around the larger numbers tend to show more as they have more combinations.

By analyzing outcomes, one can better grasp the concept of probability and make informed predictions about future events based on existing data.