GATE CSE 2021 Set 1 | GA Question: 8

Question

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.

The probability that at least two chocolates are identical is __________

• A. $0.3024$
• B. $0.4235$
• C. $0.6976$
• D. $0.8125$

Comments

why 1/10*1/9*1/8*1/7*1/6 wrong for the probability of no chocolates being identical?

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@swami_9 use basic rule of probabilty total no. of favourable outcomes/total possible outcomes

in first case we can choose any from 10 so toal no. of favourable outcomes=10 and total no. of possible outcomes=10,then in second case total no. of favourable outcomes=9 as we cannot choose the choclate that has chosen from bag 1 but total possible outcomes in this case also 10

similary for 3,4,5 bag.

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the best answer should be modified .

a case in which chocolate c1 is selected from bag B1 is different from the case in which chocolate c1 is selected from bag B2 or B3 or B4 or B5 !!

$\binom{10}{5}$ undercounts these cases , so we need to multiply it with 5! to make it equal to $_{}^{10}\textrm{}P_{5}^{}\textrm{}$

Answer 1

Option C

$P(\text{No two chocolates are identical}) = \frac{10\times9\times8\times7\times6}{{10}^5} = \frac{30240}{{10}^5} = 0.3024$

$P(\text{At least two chocolates are identical}) = 1 – P(\text{No two chocolates are identical})$

$\qquad \qquad = 1 – 0.3024 = 0.6976$

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Alternatively, 

Number of ways of selecting $5$ distinct chocolates, one each from the $5$ bags is same as selecting $5$ chocolates from $10$ distinct ones $ = {}^{10}C_5.$

If “distinct” requirement is not there, each of the $5$ chocolate has $10$ options $\implies 10^5.$

So, probability that no two chocolates are identical $ = \dfrac{{}^{10}C_5}{10^5} = 0.3024$

Probability that at least $2$ chocolcates are identical $ = 1 – 0.3024 = 0.6976$

Comments

@Pranavpurkar What is 1 chocolate is same? Does that even have a meaning? 

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OOPS!😂 , i lost my mind.

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Things that happen before D-Day. xD