Jamia Millia Islamia MCA Probability Critical Thinking Previous Year Questions (PYQs)
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Find the remainder when $67^{99}$ is divided by 7.
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Solution
$67 \equiv 4 \pmod7$, so $4^3 \equiv 1$.
$99$ is multiple of 3 → remainder $=1$.
Three letters are dictated to three persons and an envelope is addressed to each of them.
The letters are inserted into the envelopes at random so that each envelope contains exactly one letter.
What is the probability that at least one letter is in its proper envelope?
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Solution
Total permutations of 3 letters = $3! = 6$.
Derangements (no letter in correct envelope) = $!3 = 2$.
Hence, number of favorable cases (at least one correct) = $6 - 2 = 4$.
Probability $= \dfrac{4}{6} = \dfrac{2}{3}$.
A tourist visits four cities A, B, C and D in a random order.
What is the probability that he visits A before B?
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Solution
Out of all $4! = 24$ permutations, in half of them A comes before B,
and in the other half B comes before A.
Therefore probability = $\dfrac{1}{2}$.
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