Question

In an experiment, a ball is drawn from an urn containing 11 red balls and 13 purple balls. If the ball is red, two coins are tossed. Otherwise three coins are tossed.

How many elements of the sample space will have a red ball ?
How many elements of the sample space are there altogether ?

Answer 1

The sample space for drawing a red ball and tossing two coins consists of 44 elements. The total number of elements in the sample space for both red and purple balls with the associated coin tosses is 148.

Step-by-step explanation:

In the given experiment, we have an urn containing 11 red balls and 13 purple balls. When a red ball is drawn, we toss two coins, and when a purple ball is drawn, we toss three coins. To determine the number of elements in the sample space that include a red ball, we must consider the possible outcomes of tossing two coins. Each coin can land on heads (H) or tails (T), so for two coins, there are 4 possible combinations: HH, HT, TH, TT.

Since there are 11 red balls, each with the possibility of leading to 4 different coin toss outcomes, the number of elements in the sample space that include a red ball is:

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• Number of elements with a red ball = Number of red balls × Number of coin toss outcomes = 11 × 4 = 44

To find the total number of elements in the sample space, we consider both red and purple ball draws:

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• For purple balls, with 3 coin tosses, there are 2^3 = 8 possible outcomes per ball.
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• Total elements for purple balls = Number of purple balls × Number of coin toss outcomes = 13 × 8 = 104

Combining both, we get:

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• Total number of elements in the sample space = Elements with a red ball + Total elements for purple balls = 44 + 104 = 148

Answer 2

Sample space is the total number of possible outcomes. The general formula for this is r^n, where r is the number of outcomes per trial, while n is the number of trials.

For the first question, the equation would be 2² because 2 coins are tossed so n=2. Then, r=2 because there are two possible outcomes, namely, the head and the tail. So, the answer for letter a is 4.

For the second question, we will add to the answer in letter a with 2³, because this time, 3 coins are tossed. Thus, the total sample space is 4 + 2³ = 12.