An experiment is conducted for which the sample space is [tex]S=\left\{s_1, s_2, s_3, s_4, s_5\right\}[/tex]. Is the following probability assignment possible for this experiment?
[tex]
\begin{tabular}{l|ccccc}
Outcomes & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$ \\
\hline Probabilities & 0.21 & 0.06 & 0.29 & 0.25 & 0.19
\end{tabular}
[/tex]
Is the probability assignment possible for this experiment?
A. No, because not all of the probabilities are between 0 and 1, inclusive.
B. Yes, because both rules for an acceptable probability assignment are satisfied.
C. No, because the sum of the probabilities of all possible outcomes is not 1.
D. No, because none of the rules for an acceptable probability assignment are satisfied.
E. No, because the probabilities are given as decimals instead of fractions.
Answer (2)
Check if each probability is between 0 and 1: All probabilities satisfy this condition.
Calculate the sum of the probabilities: 0.21 + 0.06 + 0.29 + 0.25 + 0.19 = 1.0 .
Since both conditions are satisfied, the probability assignment is possible.
The probability assignment is possible: $\boxed{\text{Yes}}.
Explanation
Analyze the problem and data We are given a sample space S = { s 1 , s 2 , s 3 , s 4 , s 5 } and a probability assignment for each outcome. To determine if the probability assignment is possible, we need to check two conditions:
Each probability must be between 0 and 1, inclusive.
The sum of the probabilities of all possible outcomes must be equal to 1.
Check if probabilities are between 0 and 1 First, let's check if each probability is between 0 and 1:
P ( s 1 ) = 0.21 is between 0 and 1.
P ( s 2 ) = 0.06 is between 0 and 1.
P ( s 3 ) = 0.29 is between 0 and 1.
P ( s 4 ) = 0.25 is between 0 and 1.
P ( s 5 ) = 0.19 is between 0 and 1.
All probabilities are between 0 and 1, so the first condition is satisfied.
Calculate the sum of probabilities Next, let's calculate the sum of the probabilities:
P ( s 1 ) + P ( s 2 ) + P ( s 3 ) + P ( s 4 ) + P ( s 5 ) = 0.21 + 0.06 + 0.29 + 0.25 + 0.19
The sum of the probabilities is 1.0, so the second condition is satisfied.
Conclusion Since both conditions are satisfied, the probability assignment is possible for this experiment. Therefore, the correct answer is B.
Examples
Probability assignments are used in various fields such as finance, insurance, and gambling. For example, in finance, probability assignments can be used to model the likelihood of different investment outcomes. In insurance, they can be used to assess the risk of insuring different types of events. In gambling, they determine the odds of winning different games. Understanding probability assignments helps in making informed decisions in these areas.
The probability assignment is valid as all probabilities are between 0 and 1, and their total equals 1. The correct answer is B: Yes, because both rules for an acceptable probability assignment are satisfied.
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