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A deck of $5$ cards (each carrying a distinct number from $1$ to $5$) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?

1. $\left(\dfrac{1}{5}\right)$

2. $\left(\dfrac{4}{25}\right)$

3. $\left(\dfrac{1}{4}\right)$

4. $\left(\dfrac{2}{5}\right)$
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The number on the first card needs to be One higher than that on the second card, so possibilities are :

$\begin{array}{c} \begin{array}{cc} 1^{\text{st}} \text{ card} & 2^{\text{nd}} \text{ card}\\ \hline \color{red}1 & \color{red}-\\ 2 & 1\\ 3 & 2\\ 4 & 3\\ 5 & 4\\ \color{red}- & \color{red}5 \end{array}\\ \hline \text{Total$:4$possibilities} \end{array}$

Total possible ways of picking up the cards $= 5 \times 4 = 20$

Thus, the required Probability $= \dfrac{\text{favorable ways}}{\text{total possible ways}}= \dfrac{4}{20} = \dfrac 15$

Option A is correct

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How to solve the question if it would be only HIGHER and not ONE HIGHER?
+4

How to solve the question if it would be only HIGHER and not ONE HIGHER?

Possible combinations :-

 First card second card 5 4,3,2,1 4 3,2,1 3 2,1 2 1 1 none

$P(First\ card\ is\ HIGHER\ than\ second\ one\ )=\left [ \left ( \frac{1}{5} \times \frac{4}{4}\right ) + \left ( \frac{1}{5} \times \frac{3}{4}\right ) + \left ( \frac{1}{5} \times \frac{2}{4}\right ) + \left ( \frac{1}{5} \times \frac{1}{4}\right ) \right]\\ =\frac{1}{2}$

+1
Got it
Here we should consider without replacement, since "removed one at a time" means the card has been removed from the deck.

Prob of picking the first card  = 1/5

Now there are 4 cards in the deck. Prob of picking the second card = 1/4

Possible favourable combinations = 2-1, 3-2, 4-3, 5-4

Probability of each combination = (1/5)*(1/4) = 1/20

Hence answer = 4*1/20 = 1/5
with 5 cards to choose we can only fulfil the condition if we pick 2,3,4,5 in our choice else theres no way to get the same

the probability of choosing first no's is =(2,3,4,5)/(1,2,3,4,5)=4/5

the second time, we have only one option to choose out of four option=1/4

so,the total probability=(4/5)*(1/4)=1/5

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