Problem 35 - Algebra for College Students

Question

What are the odds in favor of getting exactly 2 heads in 3 tosses of a coin?

Step-by-Step Solution

Answer

The odds in favor are 3:5.

Step 1: Determine the total possible outcomes

When a coin is tossed 3 times, each toss has 2 possible outcomes: heads (H) or tails (T). Therefore, the total number of outcomes is \(2^3 = 8\). List all possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Step 2: Identify the favorable outcomes

We want to find the outcomes that have exactly 2 heads. From the list of all possible outcomes, the favorable ones are HHT, HTH, and THH. So, there are 3 favorable outcomes.

Step 3: Calculate the probability of favorable outcomes

The probability of getting exactly 2 heads in 3 tosses is the number of favorable outcomes divided by the total number of outcomes: \(\frac{3}{8}\).

Step 4: Calculate the odds in favor

The odds in favor are given by the ratio of the number of favorable outcomes to the number of unfavorable outcomes. The number of unfavorable outcomes is \(8 - 3 = 5\). Therefore, the odds in favor of getting exactly 2 heads in 3 tosses are \(\frac{3}{5}\), or 3:5.

Key Concepts

coin toss

When you toss a coin, there are two possible outcomes: heads (H) or tails (T). This is a simple example of a random experiment. If you toss the coin multiple times, each toss remains independent of the others, which means the result of one toss does not affect the others. Therefore, when you toss a coin three times, you can list all the possible outcomes as combinations of heads and tails. In this case, the total number of outcomes is given by raising the number of outcomes in one toss (which is 2) to the power of the number of tosses (which is 3).

So, we have:

• Total Outcomes = 2³ = 8
• Possible Outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Understanding this helps us proceed to the next step, which involves identifying favorable outcomes.

favorable outcomes

In probability, a favorable outcome is any result of a random experiment that satisfies the condition we are interested in. In this problem, we want to know how many times we can get exactly 2 heads in 3 tosses of a coin. We already listed all 8 possible outcomes from the first section.

Now, we go through the list and pick the outcomes that match our condition. These outcomes are HHT, HTH, and THH, which all have exactly 2 heads. So, there are 3 favorable outcomes.

We'll use this information when we calculate probability and odds. Here it is put simply:

• Total favorable outcomes = 3

odds calculation

Odds compare the likelihood of a favorable outcome to an unfavorable one. In this problem, we calculated the total number of outcomes (8) and the number of favorable outcomes (3). Now, we need to find out how many outcomes are not favorable. These are the outcomes that do not show exactly 2 heads.

This can be found by subtracting the number of favorable outcomes from the total number of outcomes:

• Unfavorable Outcomes = 8 - 3 = 5

The odds in favor are then expressed as the ratio of favorable outcomes to unfavorable outcomes:

• Odds in Favor = 3:5 or \(\frac{3}{5}\)

So, the odds in favor of getting exactly 2 heads in 3 tosses of a coin are \(\frac{3}{5}\), which means for every 3 favorable outcomes, there are 5 unfavorable ones.