1
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

The mean and variance of a binomial distribution are $$\alpha$$ and $$\frac{\alpha}{3}$$ respectively. If $$\mathrm{P}(X=1)=\frac{4}{243}$$, then $$\mathrm{P}(X=4$$ or 5$$)$$ is equal to :

A
$$\frac{5}{9}$$
B
$$\frac{64}{81}$$
C
$$\frac{16}{27}$$
D
$$\frac{145}{243}$$
2
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

Let $$\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$$ be three mutually exclusive events such that $$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}$$ and $$\mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1-\mathrm{p}}{2}$$. If the maximum and minimum values of $$\mathrm{p}$$ are $$\mathrm{p}_{1}$$ and $$\mathrm{p}_{2}$$, then $$\left(\mathrm{p}_{1}+\mathrm{p}_{2}\right)$$ is equal to :

A
$$\frac{2}{3}$$
B
$$\frac{5}{3}$$
C
$$\frac{5}{4}$$
D
1
3
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1

If $$A$$ and $$B$$ are two events such that $$P(A)=\frac{1}{3}, P(B)=\frac{1}{5}$$ and $$P(A \cup B)=\frac{1}{2}$$, then $$P\left(A \mid B^{\prime}\right)+P\left(B \mid A^{\prime}\right)$$ is equal to

A
$$\frac{3}{4}$$
B
$$\frac{5}{8}$$
C
$$\frac{5}{4}$$
D
$$\frac{7}{8}$$
4
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is :

A
$$\frac{33}{2^{32}}$$
B
$$\frac{33}{2^{29}}$$
C
$$\frac{33}{2^{28}}$$
D
$$\frac{33}{2^{27}}$$
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