Let our random experiment involve tossing two coins simultaneously. Observe the faces of the two coins. Let the sample space be given by [tex]S=\{ HH , HT , TH , TT \}[/tex], and use H to represent heads and T to represent tails. What is the probability distribution for [tex]X[/tex], where [tex]X=[/tex] The number of tails observed?
| [tex]X[/tex] | 0 | 1 | 2 |
|---|---|---|---|
| [tex]P(x)[/tex] | [tex]1 / 4[/tex] | [tex]1 / 4[/tex] | [tex]1 / 4[/tex] |
| [tex]x[/tex] | 0 | 1 |
|---|---|---|
| [tex]P(x)[/tex] | [tex]2 / 4[/tex] | [tex]1 / 4[/tex] |
| [tex]x[/tex] | 0 | 1 |
|---|---|---|
| [tex]P(x)[/tex] | [tex]1 / 4[/tex] | [tex]2 / 4[/tex] |
| [tex]x[/tex] | 0 | 1 |
|---|---|---|
| [tex]P(x)[/tex] | [tex]1 / 2[/tex] | [tex]1 / 4[/tex] |
Answer (2)
Calculate the probability of 0 tails (HH): P ( X = 0 ) = 4 1 .
Calculate the probability of 1 tail (HT, TH): P ( X = 1 ) = 4 2 = 2 1 .
Calculate the probability of 2 tails (TT): P ( X = 2 ) = 4 1 .
The probability distribution is: P ( X = 0 ) = 4 1 , P ( X = 1 ) = 4 2 , P ( X = 2 ) = 4 1 .
Explanation
Understand the problem We are given a random experiment of tossing two coins simultaneously. The sample space is S = { HH , H T , T H , TT } , where H represents heads and T represents tails. We want to find the probability distribution for X , where X is the number of tails observed.
Calculate P(X=0) First, let's find the probability of observing 0 tails, which means we observe two heads (HH). There is only one outcome in the sample space that corresponds to this event, which is HH. Since there are four equally likely outcomes in the sample space, the probability of observing 0 tails is P ( X = 0 ) = 4 1 .
Calculate P(X=1) Next, let's find the probability of observing 1 tail. This corresponds to the outcomes HT and TH. There are two outcomes in the sample space that correspond to this event. Therefore, the probability of observing 1 tail is P ( X = 1 ) = 4 2 = 2 1 .
Calculate P(X=2) Finally, let's find the probability of observing 2 tails. This corresponds to the outcome TT. There is only one outcome in the sample space that corresponds to this event. Therefore, the probability of observing 2 tails is P ( X = 2 ) = 4 1 .
Construct the probability distribution table Now, we can construct the probability distribution table for X :
\t\t\t
X 0 1 2 P ( X = x ) 4 1 4 2 4 1 \t \t \t Comparing this with the given options, none of them match exactly. However, if we consider only the values that X can take (0 and 1), we can see if any of the options match the probabilities for these values. None of the options have X taking values 0, 1 and 2. The closest option would be the one that has X taking values 0 and 1. However, the probabilities don't match our calculated probabilities. There seems to be an error in the provided options.
Final Answer However, based on our calculations, the probability distribution for X is:
\t\t\t
X 0 1 2 P ( x ) 1/4 2/4 1/4 \t \t \t Since none of the options match this distribution, we will choose the option that includes values 0 and 1 for X and has the closest probabilities. However, since none of the options include the value 2 for X, and the probabilities don't match, we can conclude that there might be an error in the provided options.
Examples
Understanding probability distributions is crucial in many real-world scenarios. For example, in quality control, we can use probability distributions to model the number of defective items in a batch. In finance, we can model the returns of an investment using probability distributions. In sports, we can model the number of goals scored in a soccer match using probability distributions. These models help us make informed decisions and predictions.
The probability distribution for the number of tails when tossing two coins shows P ( X = 0 ) = 4 1 , P ( X = 1 ) = 2 1 , P ( X = 2 ) = 4 1 . The calculated probabilities suggest that observing one tail is the most likely outcome. None of the provided options are correct based on the calculations.
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