# Aptitude - Probability - Discussion

### Discussion :: Probability - General Questions (Q.No.13)

13.

Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart, is:

[A].
 3 20
[B].
 29 34
[C].
 47 100
[D].
 13 102

Explanation:

Let S be the sample space.

 Then, n(S) = 52C2 = (52 x 51) = 1326. (2 x 1)

Let E = event of getting 1 spade and 1 heart.

 n(E) = number of ways of choosing 1 spade out of 13 and 1 heart out of 13 = (13C1 x 13C1) = (13 x 13) = 169.

 P(E) = n(E) = 169 = 13 . n(S) 1326 102

 Kavitha said: (Feb 4, 2011) Why can't it be (13C1 + 13C1) I am getting confused where 2 multiply and where 2 add can any one help me.

 Parikshit said: (May 7, 2011) The answer should be 13/51. Both of the following two cases should be considered: one is spade and the other is heart......and.. one is heart and the other is spade.... so multiply by 2..

 Probablity said: (May 8, 2011) 1st card is a spade = 13/52 AND 2nd card is a heart = 13/51 therefore 13/52 x 13/51 ... (1) OR 1st card is a heart = 13/52 AND 2nd card is a spade = 13/51 therefore 13/52 x 13/51 ... (2) Adding 1 & 2 we get 13/102

 Mdk said: (Aug 26, 2011) Thanks parikshit.

 Swetha said: (Nov 8, 2011) @kavitha. In this question, we have to find propability for taking 2 cards one is spade and another one is heart. If and comes, we should multiply likewise if or is given, then we should add.

 Xyz said: (Jan 17, 2012) @ kavita: in case of "AND" we have to "multiply" & in case of "OR" we have to "Add"

 Santosh said: (Jan 27, 2012) Here actually we discussed earlier as both are same combination we add other wise multiply. So clarify that doubt. I understood this problem but add and multiply explain.

 Amol said: (Mar 10, 2012) Thanks Probability

 Neeta said: (Oct 28, 2012) Can anybody explain me how to solve 52C2 ? or such other examples like 7C2, 13C1 etc.

 Irfan said: (Nov 7, 2012) 52C2 = 52/2*51/1 i.e. 1326 Answer. or nCr = n!/(r!)(n - r)!

 Ali said: (Mar 20, 2013) But 13/52 * 13/51 is not equal to 13/102, but rather 13/204.

 Karim said: (Jul 29, 2013) @Ali, your answer is correct. The answer should in fact be 13/204 and not 13/102. You have 13/52 for the first card and for the second it's 13/51. Multiple together you get 13/204.

 Khetesh said: (Nov 28, 2013) Actually answer 13/102 is correct. We draw 2 card: Possibility of one heart and one spade = (possibility of first spade and second heart) + (possibility of first heart and second spade). = (13/52)*(13/51) + (13/52)*(13/51). = 13*13*2/52*51. = 13/102.

 Jayalekshshmi S said: (Jul 8, 2014) Number of hearts and spades in a deck of cards is 26. Am I right?

 Vinay said: (Jul 8, 2014) Probability of A and B = P(a) * P(b). Probability of A or B = P(a) + P(b). So, 13/204 is correct, no ?

 Sulagna said: (Aug 2, 2014) If we are supposed to multiply 13/204 into 2, because w are considering 2 events= 1st spade 2nd heart and 2nd heart, 1st spade, then in the coloured marbles question, why don't we apply the same logic?

 Dhruvil said: (Aug 19, 2014) Probability = (13+13)/(52*51) = 26/(52*51) = 1/102. As there are 13 spade cards and 13 heart cards in each pack of 52 cards.

 Priya said: (Sep 7, 2014) Why can't it be 26C2/52C2?

 Shekhar said: (Sep 9, 2014) Why can't this be done ? 26P2/52C2 ? Can anyone please give me a proper explanation.

 Anamika said: (Jul 18, 2015) The probability of getting spades and hearts why not 26C52+26C51. Because spades and hearts may be from red and black colours of cards.

 Naj said: (Nov 3, 2015) Why do we have to divide (52 x 51) by 2?

 Farheen said: (Sep 25, 2016) P(c) = n(c)/n(s). = 1/52 a king of hearts . Is it true?

 Harish said: (Dec 11, 2016) Answer should be probability = 13/51.

 Aisha said: (Jan 30, 2017) From where did 13 come? Can anyone explain? please.

 Vineet Pandey said: (Jan 30, 2017) Firstly see the question. In a pack of 52 cards, there are 13 spades and 13 heart. So, now according to question. probability of getting 1 spades is =13c1 and same probability of getting heart is also 13c1 Here we write 13c1 because we have to pick a single(1) card out of total spade (13) and use same concept in heart. Now (13c1 * 13c1 )/ 52c2. So, the answer is 13/102. 52c2 is written here because there are total 52 cards out of which we pick two(2) card{ 1 of spade and 1 of heart } so that's why it is written as 52c2.

 Ashish said: (Feb 12, 2017) Thanks @Vineet.

 Uchenna said: (Jul 21, 2017) Does it mean that every pack must have 13 spade and 13 hearts?

 Vanamala Srinivas said: (Dec 8, 2017) No, the correct answer is 3/102.

 Qaisara said: (Dec 16, 2017) I am unable to understand about 13. So anyone please explain me.

 Rishit said: (May 2, 2018) @Ali and @Karim. Here 2 cards are drawn TOGETHER. Which is why the 52c2. Your answer is correct for a scenario if the question was first drawn is club and second drawn is heart or vice versa.

 Vedant said: (May 14, 2018) It is very helpful explanation, Thanks @Khetesh.

 Sasi said: (Aug 12, 2018) Why not adding two type of cards as 26c2?

 Junaidh said: (Aug 18, 2018) First card taken can be spade or heart. So the probability is 26/52. So the second card should be from the other set (spade or heart). Probability for that is 13/51. So, totally, (26/52)*(13/51) = 13/102.

 Harsha said: (Dec 8, 2018) Here how we got 13?

 Kalai said: (Aug 6, 2019) Actually answer 13/102 is correct. Because you can possible to divide it(169/1326)using by 13. 13*13 = 169. 13*102=1326, So, the answer is 13/1326.

 Sudhakar said: (Aug 23, 2019) How to get 102? The probability of cards 52C2=52*51/2*1 = 1326. The cards spade and heart 13C1*13C1 = 169. 169/1326 = 13/12.

 Satya said: (Oct 29, 2020) @Priya & @Shekhar. 26C2 would mean that you may choose any 2 cards out of 26 cards(13 spade+ 13 heart) i.e. both the cards may be the heart (for ex.) which would be wrong. The question specifically asks for 1 spade (from 13 spades) and 1 heart (from 13 heart).