CBSE 2022 · Region 5 · Set 1 · Q6 · 2 marks
Events A and B are such that \[\mathrm{P}(\mathrm{~A})=\frac{1}{2}, \mathrm{P}(\mathrm{~B})=\frac{7}{12} \text { and } \mathrm{P}(\overline{\mathrm{~A}} \cup \overline{\mathrm{~B}})=\frac{1}{4} \] Find whether the events $\displaystyle A$ and $\displaystyle \mathrm{B}$ are independent or not.A box $\displaystyle \mathrm{B}_{1}$ contains $\displaystyle 1$ white ball and $\displaystyle 3$ red balls. Another box $\displaystyle \mathrm{B}_{2}$ contains $\displaystyle 2$ white balls and $\displaystyle 3$ red balls. If one ball is drawn at random from each of the boxes $\displaystyle \mathrm{B}_{1}$ and $\displaystyle \mathrm{B}_{2}$, then find the probability that the two balls drawn are of the same colour.
Events A and B are such that \[\mathrm{P}(\mathrm{~A})=\frac{1}{2}, \mathrm{P}(\mathrm{~B})=\frac{7}{12} \text { and } \mathrm{P}(\overline{\mathrm{~A}} \cup \overline{\mathrm{~B}})=\frac{1}{4} \] Find whether the events $\displaystyle A$ and $\displaystyle \mathrm{B}$ are independent or not.
A box $\displaystyle \mathrm{B}_{1}$ contains $\displaystyle 1$ white ball and $\displaystyle 3$ red balls. Another box $\displaystyle \mathrm{B}_{2}$ contains $\displaystyle 2$ white balls and $\displaystyle 3$ red balls. If one ball is drawn at random from each of the boxes $\displaystyle \mathrm{B}_{1}$ and $\displaystyle \mathrm{B}_{2}$, then find the probability that the two balls drawn are of the same colour.
Marking-scheme solution
(a)
$\displaystyle P(A) = \dfrac{1}{2},\ P(B) = \dfrac{7}{12},\ P(\bar{A} \cup \bar{B}) = \dfrac{1}{4}$
\[P(\bar{A} \cup \bar{B}) = P(\overline{A \cap B}) = 1 - P(A \cap B)\]
\[P(A \cap B) = 1 - \frac{1}{4} = \frac{3}{4}\]
\[P(A) \times P(B) = \frac{1}{2} \times \frac{7}{12} = \frac{7}{24}\]
\[P(A) \times P(B) \neq P(A \cap B)\]
$\displaystyle \therefore$ $\displaystyle A$ and $\displaystyle B$ are not independent
Or(b) $\displaystyle P$ (both balls drawn are of same colour)
\[= P \text{ (both white) } + P \text{ (both red) }\]
\[= \frac{1}{4} \times \frac{2}{5} + \frac{3}{4} \times \frac{3}{5} = \frac{11}{20}\]
ProbabilityIndependent EventsApplyvery_short_answermedium
More from Probability
- Assertion: Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at…2023 · asked 3×
- A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances…2025 · asked 3×
- Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung…2026 · asked 3×
- The probability distribution of a random variable X is: where k is some unknown constant. The probability…2024 · asked 3×
- Recent studies suggest that roughly 12 % of the world population is left handed. Depending upon the parents,…2023 · asked 3×
- Two balls are drawn at random from a bag containing 2 red balls and 3 blue balls, without replacement. Let…2022 · asked 3×
- Let X be a random variable which assumes values x 1, x 2, x 3, x 4 such that 2 P(X=x 1)=3 P(X=x 2)=P(X=x 3)=5…2022 · asked 3×
- The probability distribution of a random variable X is given below: (i) Find the value of k. (ii) Find P(1 ≤…2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2022board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.