Problem 15 - Algebra for College Students

Question

If three coins are tossed, find the probability of getting two heads and one tail. \(\frac{3}{8}\)

Step-by-Step Solution

Answer

\(\frac{3}{8}\)

Step 1: Determine Total Possible Outcomes

When three coins are tossed, each coin has two possible outcomes: heads (H) or tails (T). Since the coin tosses are independent, we can calculate the total number of possible outcomes by multiplying the possible outcomes of each coin: \(2 \times 2 \times 2 = 8\). So, there are 8 total possible outcomes.

Step 2: Identify Favorable Outcomes for Two Heads and One Tail

We need to find the outcomes where there are exactly two heads (H) and one tail (T). The possible outcomes are HHT, HTH, and THH. Thus, there are 3 favorable outcomes.

Step 3: Calculate Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Here, the probability \(P\) of getting two heads and one tail is given by \(P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{3}{8}\).

Key Concepts

Independent Events

In probability, events are described as independent when the outcome of one event does not affect the outcome of another. When we consider coin tosses, each toss is an independent event meaning that getting a head or tail on one toss doesn't influence the result of the next toss. To illustrate, let's take a single coin being tossed multiple times. Each time you toss, there are always two outcomes: head or tail. This remains true regardless of previous results.

• If the first toss is a head, the second toss still has a 50% chance of being either a head or a tail.
• The same holds true for the third toss and any subsequent tosses.

Thus, when tossing three coins as in our exercise, even if the first two show heads, the result of the third coin is totally independent of the first two. This principle of independence helps us compute the total possible outcomes by considering each toss separately.

Favorable Outcomes

A favorable outcome is essentially any result that aligns with the specifics of the situation or event that you’re analyzing. In our exercise, we want two heads and one tail when tossing three coins. With three coin tosses, we can actually have several sequences that match this criteria:

• Head, Head, Tail (HHT)
• Head, Tail, Head (HTH)
• Tail, Head, Head (THH)

Each of these sequences is considered a favorable outcome because they meet our condition of exactly two heads and one tail. Understanding which outcomes are favorable is crucial for determining probability, as it directly influences the calculation by providing the numerator in the probability formula.

Total Possible Outcomes

The total possible outcomes pertain to all the potential results that might occur in a probability scenario. For coin tosses, every toss is an independent event with two outcomes: heads or tails. When we toss three coins, each coin's two outcomes multiply across the number of coins tossed: \[ 2 \times 2 \times 2 = 8 \] Thus, there are 8 total possible outcomes when tossing three coins.

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

Each sequence represents a unique arrangement of heads and tails. This set of possible outcomes forms the denominator in our probability calculations, confirming the framework upon which the entire probability is based.