Given that events A and B are mutually exclusive, so the probability of either event occurring at the same time is zero. Also, the sum of probabilities for mutually exclusive events should be less than or equal to 1 . This gives us the inequality:
P(A)+P(B)≤1 We are given the probabilities:
‌P(A)=‌(3x+1)‌P(B)=‌(1−x) Substituting these into our inequality:
‌(3x+1)+‌(1−x)≤1To simplify, find a common denominator, which is 12 :
‌+‌≤1 Combine the fractions:
‌≤1 Expand and combine like terms:
‌‌≤1‌‌≤1 Multiply both sides by 12 to clear the denominator:
9x+7≤12Subtract 7 from both sides:
9x≤5Divide both sides by 9 :
x≤‌ Also, since probabilities must be non-negative:
P(A)≥0‌‌‌ and ‌‌‌P(B)≥0So:
‌(3x+1)≥0Multiply both sides by 3 :
3x+1≥0Subtract 1 from both sides:
3x≥−1Divide by 3 :
x≥−‌And for
P(B) :
‌(1−x)≥0 Multiply both sides by 4 :
1−x≥0Subtract 1 from both sides:
−x≥−1Multiply by -1 (which reverses the inequality):
x≤1 Combining these results:
−‌≤x≤‌Therefore, the possible values of
x lie in the interval:
Option B:
[−‌,‌]