question archive A box contains 11 cards numbered from 1 to 11 and 5 face cards

A box contains 11 cards numbered from 1 to 11 and 5 face cards

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A box contains 11 cards numbered from 1 to 11 and 5 face cards. Two cards are randomly drawn from the box without replacement. Find the probability

 

a) P(both are face cards)

 

b) P(both cards are even number)

 

c) P(a face card and an odd number)

 

2) : In a classroom with 15 students, how many different groups of 4 students can be formed?

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1) a)0.0833

b)0.0833

c)0.25

 

2)1365

Step-by-step explanation

Q1) We have 16 cards of which 11 are numbered 1 to 11 and 5 are face cards

 

a) To find P(both are face cards)

During the first draw, there are 5 face cards among 16 cards.

So probability of obtaining a face card on first draw = 5/16

During the second draw, there are only 4 face cards left out of a total of 15 cards (Since we have to find probability of drawing 2 face cards, we are assuming we already picked one card which is a face card, before making the second draw)

So probability of obtaining a face card on second draw assuming he drew a face in first draw = 4/15

 

SO, P(both are face cards) = (5/16) x (4/15) = 1/12 = 0.0833

 

b) P(both cards are even number)

During the first draw, there are 5 even number cards [2, 4, 6, 8, 10] among 16 cards.

So probability of obtaining an even card on first draw = 5/16

During the second draw, there are only 4 even cards left out of a total of 15 cards

So probability of obtaining an even card on second draw assuming he drew an even card in first draw = 4/15

 

SO, P(both cards are even number) = (5/16) x (4/15) = 1/12 = 0.0833

 

c) To find P(a face card and an odd number)

Here there are two ways to pick the cards. First a face and then an odd number OR first an odd number and then a face card:

 

Drawing face card first:

 

During the first draw, there are 5 face cards among 16 cards.

So probability of obtaining a face card on first draw = 5/16

During the second draw, there are 6 odd cards [1,3,5,7,9,11] out of a total of 15 cards

So probability of obtaining an odd card on second draw assuming he drew a face card in first draw = 6/15 = 2/5

So P(a face card on first draw and an odd number on 2nd draw) = (5/16) x (2/5) = 1/8 = 0.125

 

Drawing odd card first:

 

During the first draw, there are 6 odd cards among 16 cards.

So probability of obtaining an odd card on first draw = 6/16 = 3/8

During the second draw, there are 5 face cards out of a total of 15 cards

So probability of obtaining a face card on second draw assuming he drew an odd card in first draw = 5/15 = 1/3

So P(an odd number on first draw and a face card on 2nd draw) = (3/8) x (1/3) = 1/8 = 0.125

 

So, total probability,

P(a face card and an odd number) = 1/8 + 1/8 = 2/8 = 1/4 = 0.25

 

2) Given, there are 15 students.

We need to choose group of 4 students from the 15 students.

Here we can use combination operator. (Permutation is not required because here order is not relevant)

 

Hence total number of ways to make groups of 4 = 15C4 = ?(15−4)!×4!15!?? = ?11!×4!15!?? = ?1×2×3×412×13×14×15?? = 1365

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