Respuesta :
Answer:
Probability of diamond card selected when 5 is write down = [tex]\frac{2}{11}[/tex]
Step-by-step 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) = [tex]\frac{13}{52}[/tex] = [tex]\frac{1}{4}[/tex]
P(not a diamond card) = 1 - [tex]\frac{1}{4}[/tex] = [tex]\frac{3}{4}[/tex]
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 ) = [tex]\frac{4}{36}[/tex] = [tex]\frac{1}{9}[/tex]
P(Sum 5 when one dice is rolled ) = [tex]\frac{1}{6}[/tex]
Now,
P(sum 5 when diamond card is selected) = [tex]\frac{1}{4}[/tex] × [tex]\frac{1}{9}[/tex] = [tex]\frac{1}{36}[/tex]
P(sum 5 when diamond card is not selected = [tex]\frac{3}{4}[/tex] × [tex]\frac{1}{6}[/tex] = [tex]\frac{1}{8}[/tex]
Now,
Probability (Getting 5) = [tex]\frac{1}{36}[/tex] + [tex]\frac{1}{8}[/tex] = [tex]\frac{11}{72}[/tex]
Now,
Probability of diamond card selected when 5 is write down = [tex]\frac{\frac{1}{36} }{\frac{11}{72} }[/tex]
= [tex]\frac{1}{36}[/tex] ×[tex]\frac{72}{11}[/tex] = [tex]\frac{2}{11}[/tex]
⇒Probability of diamond card selected when 5 is write down = [tex]\frac{2}{11}[/tex]
