Question

a) Draw a tree diagram that depicts tossing a coin three times. Use H to represent a head and T to represent a tail landing face up. List the arrangements of heads (H) and tails (T) by the branches of your tree diagram. b) Expand \((\mathrm{H}+\mathrm{T})^{3}\) by multiplying the factors. In the first step write the factors in full. For example, the first term will be HHH. You should have eight different terms. Simplify this arrangement of terms by writing HHH as \(\mathrm{H}^{3},\) and so on. Combine like terms. c) What does HHH or \(\mathrm{H}^{3}\) represent in both part a) and part b)? Explain what 3HHT or \(3 \mathrm{H}^{2} \mathrm{T}\) represents in parts a) and b).

Short answer

The outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. The expansion of \(H + T)^3\ is \(H^3 + 3H^2T + 3HT^2 + T^3\. \(H^3\ represents three heads, \(3H^2T\ represents two heads and one tail.

Step-by-step solution

Draw the tree diagram for tossing a coin three times

Start with a single point and draw two branches originating from it, one labeled H (for heads) and the other labeled T (for tails). From each of these branches, draw two more branches for the second toss, and label them H and T. Repeat this process one more time for the third toss. This will give you a complete tree diagram showing all possible outcomes of tossing a coin three times. The final outcomes at the end of the tree will be: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

List the arrangements

List all the final outcomes obtained from the tree diagram: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Expand \((H + T)^3\)

Write the expanded form by multiplying the factors. First, write \((H + T) (H + T) (H + T)\). The expansion is done by choosing one term from each factor in all possible ways: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Combine like terms

Group and simplify similar terms using exponential notation: \(HHH = H^3, \ HHT, HTH, THH = 3H^2T, \ HTT, THT, TTH = 3HT^2, \ TTT = T^3\). Thus, the simplified form of the expansion is \(H^3 + 3H^2T + 3HT^2 + T^3\).

Interpret the terms

In part (a), HHH represents the event of getting heads in all three tosses. In part (b), \(H^3\) represents the probability of getting heads three times—since it only appears once, it stands for 1 way to get HHH. Similarly, \(3H^2T\) in both parts represents getting exactly two heads and one tail in any order, which can happen in three different ways (HHT, HTH, THH).

Key concepts

Coin Toss Probability

When we talk about coin tosses, probability helps us understand the chances of getting heads (H) or tails (T). Each toss has two possible outcomes: H or T, with equal probabilities of 1/2. To visualize this, we can use a tree diagram.

Here's how it works:

• Start with a single point for the first toss, branching into H and T.
• For the second toss, each branch splits again into H and T.
• This continues for the third toss.

After drawing the diagram, you will see all possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

This tree diagram helps to list all outcomes in an organized way. It also shows that each outcome has the same probability of occurring.

Binomial Expansion

Binomial expansion is a method used to expand expressions raised to a power. In this exercise, we expand \((H + T)^3\). Here's how to do it step-by-step:

1. Start with \((H + T) (H + T) (H + T)\).
2. Multiply the factors by choosing one term from each factor in all possible ways.

The expanded terms for three coin tosses are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Next, we combine and simplify these terms using exponential notation:

• HHH = H³
• HHT, HTH, and THH = 3H²T
• HTT, THT, and TTH = 3HT²
• TTT = T³

Thus, the simplified binomial expansion is \ H^3 + 3H^2T + 3HT^2 + T^3 \.

Combinatorics

Combinatorics deals with counting arrangements and combinations. Let's dive into the arrangements shown in our binomial expansion and tree diagram.

Each term in the expansion represents different combinations and frequencies of heads (H) and tails (T). For example:

• \( H^3 \) or HHH represents getting heads in all three tosses.
• \( 3H^2T \) or three terms like HHT, HTH, THH represent two heads and one tail in any order.
• \( 3HT^2 \) stands for outcomes like HTT, THT, TTH where one head and two tails occur.
• \( T^3 \) or TTT is the scenario where all tosses result in tails.

In summary, combinatorics helps us count and understand the number of ways each outcome can be arranged. These principles are useful not only for coin tosses, but also in many other probability problems and real-life scenarios.