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If (1+3p)/(3),(1-p)/(4) and (1-2p)/(2) a...

If `(1+3p)/(3),(1-p)/(4) and (1-2p)/(2)` are the probabilities of three mutually exclusive events, then the set of all values of p is _____.

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To find the set of all values of \( p \) for which the expressions \( \frac{1 + 3p}{3} \), \( \frac{1 - p}{4} \), and \( \frac{1 - 2p}{2} \) are valid probabilities of mutually exclusive events, we need to ensure that each probability lies within the range [0, 1]. ### Step-by-step Solution: 1. **Define the probabilities**: - Let \( P(E_1) = \frac{1 + 3p}{3} \) - Let \( P(E_2) = \frac{1 - p}{4} \) - Let \( P(E_3) = \frac{1 - 2p}{2} \) 2. **Set up inequalities for each probability**: - For \( P(E_1) \): \[ 0 \leq \frac{1 + 3p}{3} \leq 1 \] - For \( P(E_2) \): \[ 0 \leq \frac{1 - p}{4} \leq 1 \] - For \( P(E_3) \): \[ 0 \leq \frac{1 - 2p}{2} \leq 1 \] 3. **Solve the inequalities for \( P(E_1) \)**: - From \( 0 \leq \frac{1 + 3p}{3} \): \[ 1 + 3p \geq 0 \implies 3p \geq -1 \implies p \geq -\frac{1}{3} \] - From \( \frac{1 + 3p}{3} \leq 1 \): \[ 1 + 3p \leq 3 \implies 3p \leq 2 \implies p \leq \frac{2}{3} \] - Thus, for \( P(E_1) \): \[ -\frac{1}{3} \leq p \leq \frac{2}{3} \] 4. **Solve the inequalities for \( P(E_2) \)**: - From \( 0 \leq \frac{1 - p}{4} \): \[ 1 - p \geq 0 \implies p \leq 1 \] - From \( \frac{1 - p}{4} \leq 1 \): \[ 1 - p \leq 4 \implies -p \leq 3 \implies p \geq -3 \] - Thus, for \( P(E_2) \): \[ -3 \leq p \leq 1 \] 5. **Solve the inequalities for \( P(E_3) \)**: - From \( 0 \leq \frac{1 - 2p}{2} \): \[ 1 - 2p \geq 0 \implies 2p \leq 1 \implies p \leq \frac{1}{2} \] - From \( \frac{1 - 2p}{2} \leq 1 \): \[ 1 - 2p \leq 2 \implies -2p \leq 1 \implies p \geq -\frac{1}{2} \] - Thus, for \( P(E_3) \): \[ -\frac{1}{2} \leq p \leq \frac{1}{2} \] 6. **Combine the results**: - From \( P(E_1) \): \( -\frac{1}{3} \leq p \leq \frac{2}{3} \) - From \( P(E_2) \): \( -3 \leq p \leq 1 \) - From \( P(E_3) \): \( -\frac{1}{2} \leq p \leq \frac{1}{2} \) The common interval is: \[ -\frac{1}{2} \leq p \leq \frac{1}{2} \] ### Final Answer: The set of all values of \( p \) is: \[ \boxed{[-\frac{1}{2}, \frac{1}{2}]} \]
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ML KHANNA-PROBABILITY-Problem Set (1) FILL IN THE BLANKS
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  12. If ((1-3p))/2,((1+4p))/3,((1+p))/6 are the probabilities of three m...

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  13. If E(1) and E(2) are two events and E(2) is a subset of E(1) then P(E(...

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  14. If two events A and B are such that P(overline(A))=0.3, P(B)=0.4 and P...

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  15. If P(AcupB)=0*9,P(B)=0*4 when A and B are independent events then P(A)...

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  16. Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 ...

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