CBSE 2025 · Region 4 · Set 3 · Q29 · 3 marks
The probability distribution of a random variable X is given by : X $\displaystyle 0$ $\displaystyle 1$ $\displaystyle 2$ $\displaystyle 3$ $\displaystyle \mathrm{P}(\mathrm{X})$ p $\displaystyle \frac{\mathrm{p}}{3}$ $\displaystyle \frac{\mathrm{p}}{6}$ $\displaystyle \frac{\mathrm{p}}{12}$
(i)Determine the value of p .(ii)Calculate $\displaystyle \mathrm{P}(\mathrm{X} \geq 1)$.(iii)Calculate expectation of X , i.e. $\displaystyle \mathrm{E}(\mathrm{X})$In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that $\displaystyle 50 \%$ people preferred using public transport, $\displaystyle 35 \%$ preferred using a bicycle and $\displaystyle 20 \%$ use both public transport and a bicycle. If a person is selected at random, find the probability that:(i)The person uses only public transport.(ii)The person uses a bicycle, given that they also use the public transport.(iii)The person uses neither public transport nor a bicycle.
The probability distribution of a random variable X is given by :
| X | $\displaystyle 0$ | $\displaystyle 1$ | $\displaystyle 2$ | $\displaystyle 3$ |
| $\displaystyle \mathrm{P}(\mathrm{X})$ | p | $\displaystyle \frac{\mathrm{p}}{3}$ | $\displaystyle \frac{\mathrm{p}}{6}$ | $\displaystyle \frac{\mathrm{p}}{12}$ |
(i)
Determine the value of p .
(ii)
Calculate $\displaystyle \mathrm{P}(\mathrm{X} \geq 1)$.
(iii)
Calculate expectation of X , i.e. $\displaystyle \mathrm{E}(\mathrm{X})$
In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that $\displaystyle 50 \%$ people preferred using public transport, $\displaystyle 35 \%$ preferred using a bicycle and $\displaystyle 20 \%$ use both public transport and a bicycle. If a person is selected at random, find the probability that:
(i)
The person uses only public transport.
(ii)
The person uses a bicycle, given that they also use the public transport.
(iii)
The person uses neither public transport nor a bicycle.
Marking-scheme solution
\[\begin{aligned}
& \text { (i) } \mathrm{p}+\frac{\mathrm{p}}{3}+\frac{\mathrm{p}}{6}+\frac{\mathrm{p}}{12}=1 \\
& \Rightarrow \mathrm{p}=\frac{12}{19}
\end{aligned}
\]
Let $\displaystyle T$ :Person uses public transport
$\displaystyle B$ :Person uses a bicycle
Given $\displaystyle \mathrm{P}(T)=\frac{50}{100}, \mathrm{P}(B)=\frac{35}{100}, \mathrm{P}(T \cap B)=\frac{20}{100}$
(i)
$\displaystyle \mathrm{P}($ only $\displaystyle T)=\mathrm{P}(T)-\mathrm{P}(T \bigcap B)$
\[=\frac{50}{100}-\frac{20}{100}=\frac{30}{100}
\]
(ii)
$\displaystyle \mathrm{P}(B \mid T)=\frac{\mathrm{P}(T \bigcap B)}{\mathrm{P}(T)}$
\[=\frac{\dfrac{20}{100}}{\dfrac{50}{100}}=\frac{2}{5}
\]
(iii)
$\displaystyle \mathrm{P}\left(T^{\prime} \cap B^{\prime}\right)=1-\mathrm{P}(T \bigcup B)$
\[\begin{aligned}
& =1-[\mathrm{P}(T)+\mathrm{P}(B)-\mathrm{P}(T \cap B)] \\
& =1-\left[\frac{50}{100}+\frac{35}{100}-\frac{20}{100}\right] \\
& =1-\frac{65}{100}=\frac{35}{100}
\end{aligned}
\]
ProbabilityRandom Variable and its Probability DistributionApplyshort_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 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.