Let a sample space be
S={ω1,ω2,…,ω6}
. Which of the following assignments of probabilities to each outcome are valid?
Outcomes ω1ω2ω3ω4ω5ω6 (a) 616161616161 (b) 100000 (c) 81323131−41−31 (d) 12112161616123 (e) 0.10.20.30.40.50.6
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Step-by-Step Solution
Step 1: Understand Probability Rules
For any assignment of probabilities to be valid, two fundamental rules must be satisfied. First, the probability of each individual outcome,
P(ωi)
, must be between 0 and 1, inclusive. Second, the sum of the probabilities of all possible outcomes in the sample space must equal 1.
Step 2: Check Assignment (a)
For assignment (a), all probabilities are
1/6
, which is between 0 and 1. The sum of these probabilities is
6×(1/6)=1
. Both conditions are met, so this is a valid assignment.
Step 3: Check Assignment (b)
For assignment (b), all probabilities are either 0 or 1, which are within the valid range. The sum of these probabilities is
1+0+0+0+0+0=1
. Both conditions are met, so this is a valid assignment.
Step 4: Check Assignment (c)
In assignment (c), the probabilities
P(ω5)=−1/4
and
P(ω6)=−1/3
are negative. Probabilities cannot be negative. Therefore, this is not a valid assignment.
Step 5: Check Assignment (d)
For assignment (d), the probability
P(ω6)=3/2=1.5
is greater than 1. Probabilities cannot exceed 1. Therefore, this is not a valid assignment.
Step 6: Check Assignment (e)
For assignment (e), all individual probabilities are between 0 and 1. However, the sum of the probabilities is
0.1+0.2+0.3+0.4+0.5+0.6=2.1
. Since the sum is not equal to 1, this is not a valid assignment.