Consider the experiment of simultaneously tossing a fair coin and rolling a fair die. Let X denote the number of heads showing on the coin and Y the number of spots showing on the die.
a. List the outcomes in S.
b. Find Fx,y(1,2).
Part a is easy. It's part b that I don't understand. The answer given in the book is 4/12 but I don't see how they got that.
If you got part (a) then you know all possible outcomes.
How many of those show 1 head and 2 spots?
I don't see how 4/12 is the answer either, unless F(1,2) means at least 2 spots. Even that would be 5/12.
To find the answer to part b, we need to understand what Fx,y(1,2) represents.
Fx,y(1,2) denotes the probability that the random variables X and Y take the values 1 and 2 respectively.
To calculate this probability, we need to determine the total number of favorable outcomes (where the coin shows 1 head and the die shows 2 spots) and divide it by the total number of possible outcomes.
Let's break down the problem step by step:
Step 1: Determine the sample space S.
In this experiment, we simultaneously toss a coin and roll a die. The possible outcomes for a coin toss are either a head (H) or a tail (T), and the possible outcomes for rolling a die are numbers from 1 to 6.
So the sample space S consists of all possible outcomes when both the coin and the die are tossed. The sample space S can be represented as the set of ordered pairs {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}.
Step 2: Determine the favorable outcomes for Fx,y(1,2).
We are interested in the outcome where X = 1 (1 head) and Y = 2 (2 spots on the die).
From the sample space S, we can see that there is only one outcome that satisfies these conditions: (H, 2).
Step 3: Determine the total number of outcomes.
Since we have equally likely outcomes for both the coin toss and the die roll, the total number of outcomes is the product of the number of outcomes for each event. In this case, we have 2 possible outcomes for the coin (H, T) and 6 possible outcomes for the die (1, 2, 3, 4, 5, 6). Therefore, the total number of outcomes is 2 * 6 = 12.
Step 4: Calculate the probability.
Now that we have determined the number of favorable outcomes and the total number of outcomes, we can calculate the probability using the formula:
P(X = 1, Y = 2) = Number of favorable outcomes / Total number of outcomes.
In this case, P(X = 1, Y = 2) = 1/12.
So the answer given in the book of 4/12 is incorrect if we are looking for Fx,y(1,2). The correct answer should be 1/12.