Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?
Outcome Probability $\omega_{1}$ $1/8$ $\omega_{2}$ $2/3$ $\omega_{3}$ $1/3$ $\omega_{4}$ $1/3$ $\omega_{5}$ $-1/4$ $\omega_{6}$ $-1/3$
| Outcome | Probability |
|---|---|
| $\omega_{1}$ | $1/8$ |
| $\omega_{2}$ | $2/3$ |
| $\omega_{3}$ | $1/3$ |
| $\omega_{4}$ | $1/3$ |
| $\omega_{5}$ | $-1/4$ |
| $\omega_{6}$ | $-1/3$ |
(NONE) For an assignment of probabilities to be valid,it must satisfy two conditions:
$1$. Each probability $P(\omega_{i})$ must be such that $0 \le P(\omega_{i}) \le 1$ for all $i$.
$2$. The sum of all probabilities must be equal to $1$,i.e.,$\sum_{i=1}^{6} P(\omega_{i}) = 1$.
In the given assignment,we observe that $P(\omega_{5}) = -1/4$ and $P(\omega_{6}) = -1/3$.
Since these probabilities are negative,they violate the first condition $(0 \le P(\omega_{i}) \le 1)$.
Therefore,this assignment of probabilities is not valid.
$1$. Each probability $P(\omega_{i})$ must be such that $0 \le P(\omega_{i}) \le 1$ for all $i$.
$2$. The sum of all probabilities must be equal to $1$,i.e.,$\sum_{i=1}^{6} P(\omega_{i}) = 1$.
In the given assignment,we observe that $P(\omega_{5}) = -1/4$ and $P(\omega_{6}) = -1/3$.
Since these probabilities are negative,they violate the first condition $(0 \le P(\omega_{i}) \le 1)$.
Therefore,this assignment of probabilities is not valid.
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