Question

You decide to perform an experiment with a standard deck of 52 cards and a set of 6-sideddice. You start by randomly picking a card from the deck. If the card is a diamond, you roll2 dice and record the sum of the dice. Otherwise, you roll 1 die and record the value of thedie. If you wrote down a 5, what is the probability the card you drew was a diamond

Answer 1

Probability of diamond card selected when 5 is write down =
`(2)/(11)`

Explanation:

Given - You decide to perform an experiment with a standard deck of 52 cards and a set of 6-sided dice. You start by randomly picking a card from the deck. If the card is a diamond, you roll 2 dice and record the sum of the dice. Otherwise, you roll 1 die and record the value of the die.

To find - If you wrote down a 5, what is the probability the card you drew was a diamond ?

Proof -

Total Number of cards in a deck = 52

Total number of Diamond card in a deck = 13

Now,

P(diamond card) =
`(13)/(52)` =
`(1)/(4)`

P(not a diamond card) = 1 -
`(1)/(4)` =
`(3)/(4)`

Now,

Total outcomes when two dice are rolled = 36

Total outcome of getting 5 when two dice are rolled = { (1,4), (2,3). (3,2), (4,1) }

So,

P(Sum 5 when two dice are rolled ) =
`(4)/(36)` =
`(1)/(9)`

P(Sum 5 when one dice is rolled ) =
`(1)/(6)`

Now,

P(sum 5 when diamond card is selected) =
`(1)/(4)` ×
`(1)/(9)` =
`(1)/(36)`

P(sum 5 when diamond card is not selected =
`(3)/(4)` ×
`(1)/(6)` =
`(1)/(8)`

Now,

Probability (Getting 5) =
`(1)/(36)` +
`(1)/(8)` =
`(11)/(72)`

Now,

Probability of diamond card selected when 5 is write down =
`((1)/(36) )/((11)/(72) )`

=
`(1)/(36)` ×
`(72)/(11)` =
`(2)/(11)`

⇒Probability of diamond card selected when 5 is write down =
`(2)/(11)`