Probability Quiz-1

Solution:
Out of a total of 6 occurrences, there is only one possibility of getting 4 as on the dice

so the required probability will be 1/6

Solution:
In a pack of card, there are a total of 13 spades card,

so the chances for being it a spade card will be
=13/52
=1/4

Solution:
Atleast one tail is the non-event for all heads

Thus, P(at least 1 tail) = 1-P(all heads)

=1-\frac{1}{8} = \frac{7}{8}

Solution:
Odds against an event = \frac{p(E')}{p(E)}

In this case, the event is All black, i.e; First is black and second is black and third is black

P(E) = \frac{5}{9}\times \frac{4}{8}\times \frac{3}{7}= \frac{60}{504} = 5\frac{5}{42}

Odds against the event  = \frac{37}{5}

Solution:
The event definition would be given by

first is blue and second is blue = \frac{3}{11}* \frac{2}{10} = \frac{3}{55}

Solution:
Probability of drawing a king,queen and jack is=3!\times \frac{4}{52}\times \frac{4}{51} \times \frac{4}{50}=\frac{16}{5525}

Solution:
P(E1) = \frac{3}{8}

P(E2) = \frac{7}{12}

Event definition is : E1 occurs and E2 does not occur or E1 occurs and E2 occurs or E2 occurs and E1 does not occur.

(\frac{3}{8}\times \frac{5}{12})+(\frac{3}{8}\times \frac{7}{12})+(\frac{5}{8}\times \frac{7}{12}) = \frac{71}{96}

Solution:
the possible outcomes are:

(1,1); (1,2) ; (2,1) ; (2,2) ; (3,1) ; (1,3)

Out of six cases in two cases there is exaclty one '2'

Thus the correct answer is 2/6 = 1/3

Solution:
They will contradict each other if Rajesh is true and Aman is false or vice-versa

(\frac{3}{8})\times(\frac{1}{6})+(\frac{5}{8})\times (\frac{5}{6})
= \frac{28}{48} = \frac{7}{12}

Solution:
_{2}C^{4}\textrm{} \times (\frac{6}{36})^{2} \times (\frac{30}{36})^{2} = 6\times \frac{1}{36}\times \frac{25}{36} = \frac{25}{216}