Problem 9 - Algebra for College Students

Question

If a pair of coins is tossed, then what is the probability of getting a) exactly two heads? b) at least one tail? c) exactly two tails? d) at most one tail?

Step-by-Step Solution

Answer

a) \(\frac{1}{4}\)b) \(\frac{3}{4}\)c) \(\frac{1}{4}\)d) \(\frac{1}{2}\)

Step 1: Identify the Sample Space

When two coins are tossed, the possible outcomes (sample space) are: HH, HT, TH, TT. This gives us a total of 4 outcomes.

Step 2: Calculate Probability for Exactly Two Heads

There is only one outcome with exactly two heads, which is HH. Hence, the probability is given by \(\frac{1}{4}\).

Step 3: Calculate Probability for at Least One Tail

The outcomes with at least one tail are HT, TH, and TT. Therefore, the probability is \(\frac{3}{4}\).

Step 4: Calculate Probability for Exactly Two Tails

There is only one outcome with exactly two tails, which is TT. Hence, the probability is \(\frac{1}{4}\).

Step 5: Calculate Probability for at Most One Tail

The outcomes with at most one tail are HH and HT. Therefore, the probability is \(\frac{1}{2}\).

Key Concepts

sample space

In probability, the sample space is the set of all possible outcomes of an experiment. When you toss two coins, each coin has two possibilities: heads (H) or tails (T). So, the sample space for two coin tosses includes all combinations of these outcomes. The possible outcomes are:

HH, HT, TH, TT

This gives us a total of four outcomes. Understanding the sample space is crucial because it forms the basis for calculating probabilities in any experiment. Knowing all possible outcomes helps you determine how many of these outcomes meet the criteria of the event you're interested in.

probability calculation

To calculate probability, we use the formula:
\(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)

Let’s apply this formula to solve the problems from the exercise:

a) **Exactly two heads:**
There is only one favorable outcome (HH) out of the four possible outcomes. Thus, the probability is: \(\frac{1}{4}\).

b) **At least one tail:**
The outcomes that meet this condition are HT, TH, and TT, giving us three favorable outcomes out of four. So, the probability is: \(\frac{3}{4}\).

c) **Exactly two tails:**
There is only one favorable outcome (TT) out of the four possible outcomes, making the probability: \(\frac{1}{4}\).

d) **At most one tail:**
The outcomes meeting this condition are HH and HT, providing us with two favorable outcomes out of four. Therefore, the probability is: \(\frac{1}{2}\).

combinatorics

Combinatorics is the branch of mathematics that deals with counting combinations of objects. It plays a significant role in probability and helps to figure out how many ways an event can occur.

When tossing a pair of coins, combinatorics helps us understand the different outcomes:

* Each coin has 2 outcomes.
* For two coins, you calculate the total possible outcomes by multiplying the number of outcomes for each coin: \[2 \times 2 = 4\]

These are denoted as:
* HH (both heads)
* HT (first heads, second tails)
* TH (first tails, second heads)
* TT (both tails)

By counting these combinations, we easily determine the total number of outcomes and can then apply probability principles to any specific event.

independent events

In probability theory, two events are independent if the outcome of one does not affect the outcome of the other. Tossing coins is a classic example of independent events.

When you toss two coins, the result of the first coin does not influence the result of the second coin. Each coin has 2 possible outcomes (H or T), and these are independent of each other.

Understanding independence is important because it simplifies calculations and helps us to break down complex problems into simpler parts. In our exercise, the probability calculations for each desired outcome hinge on the independence of the coin tosses.