Problem 17 - Precalculus with Limits

Question

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting a head on the first toss

Step-by-Step Solution

Answer

The probability of getting a head on the first toss is \( \frac{1}{2} \) or 0.5.

Step 1: Define the sample space

The sample space, \( S \), defines all the possible outcomes of an experiment. In this case, \( S \) denotes the outcomes of tossing a coin three times, which is given as \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \). Each symbol in the outcomes denotes the result of a toss where 'H' stands for Heads and 'T' for Tails.

Step 2: Identify the favourable outcomes

We are required to find the probability of getting a head on the first toss. So 'HTH', 'HHT', 'HHT' and 'HHH' are the favourable outcomes since in all these, the first toss gives a head.

Step 3: Use the formula of Probability

The probability of an event is the ratio of the favourable outcomes to the total number of outcomes in the sample space. Since there are 4 outcomes where the first toss is Head and 8 outcomes in total, the probability, \( P(H) \), of getting a head on the first toss is given by the formula \( P(H) = \frac{Number\,of\,favourable\,outcomes}{Total\,number\,of\,outcomes} = \frac{4}{8} \).

Key Concepts

Sample Space

The concept of a sample space is crucial in probability as it encompasses all of the possible outcomes of an experiment. For example, when tossing a coin three times, each sequence of heads (H) and tails (T) is considered an outcome. In this case, the sample space, denoted by \( S \), includes the outcomes \( \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \). Here, each element represents one possible result from three coin tosses, where "H" means a Head and "T" means a Tail.

Understanding sample space is the first step in tackling any probability problem as it provides the total set of possibilities from which we can determine specific events. By laying out all potential outcomes, we can easily analyze the specific scenarios and determine the probability of each event.

Favourable Outcomes

Favourable outcomes are specific outcomes of an experiment that satisfy the condition or the event we are interested in. In our exercise, we are focused on the event of getting a head on the first toss of the coin.

To determine which outcomes from the sample space are favourable, we look only at those sequences where the first toss results in a head. From the sample space \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \), the outcomes that fit this event are \( \{HHH, HHT, HTH, HTT\} \). These are the specific scenarios where the first coin lands on a head.

• HHH: Head on all three tosses.
• HHT: Head on the first toss, followed by two tails.
• HTH: Head on the first toss, followed by tail and head.
• HTT: Head on the first toss, followed by two tails.

Identifying favourable outcomes helps us refine the probability calculation by focusing exactly on those cases that meet our event criteria.

Probability Formula

The probability formula is a fundamental principle in probability theory. It helps determine how likely a specific event is to occur. The formula is expressed as:\[P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes in the sample space}}\]For the given task of finding the probability of getting a head on the first toss, we identified 4 favourable outcomes (HHH, HHT, HTH, HTT) from a total of 8 possible outcomes. Plugging this into our formula gives:\[P(H) = \frac{4}{8} = \frac{1}{2}\]This calculation shows that there is a 50% chance, or probability of \( \frac{1}{2} \), of getting a head on the first toss when a coin is tossed three times.

Using the probability formula allows us to make informed predictions about future occurrences based on past data or theoretical possibilities.