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Mathematics · 2025 · 3 marks
CBSE 2025 · Region 4 · Set 3 · Q29
(a) The probability distribution of a random variable X is given by: (i) Determine the value of p. (ii) Calculate P ( X ≥ 1). (iii) Calculate…
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)
$\displaystyle \mathrm{P}\left(T^{\prime} \cap B^{\prime}\right)=1-\mathrm{P}(T \bigcup B)$
\[\begin{aligned}
& =1-[\mathrm{P}(T)+\mathrm{P}(B)-\mathrm{P}(T \cap B)] \\
& =1-\left[\frac{50}{100}+\frac{35}{100}-\frac{20}{100}\right] \\
& =1-\frac{65}{100}=\frac{35}{100}
\end{aligned}
\]
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CBSE Class 12 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.