Answer to Question #265529 in Statistics and Probability for Maurice

Question

Rolling a dice three times, evaluate the probability of having at least one 6.

Drawn a card from a deck of 52 cards, verify whether the following events are statistically independent:

a) A = {drawing of a picture card}; B ={drawing of a hearth card} 

b) What if the king of hearths is missing from the deck of cards? 

c) What if a card, at random, is missing? (5 marks)

d) if a card is drawn at random from a deck of cards. Find the probability of getting the King of hearts.

Answer 1

Solution:

The probability of having at least one 6 = (Getting one 6 on one toss, not getting 6 on two tosses) + (getting two 6 on two toss, not getting 6 on one toss) + (getting three 6 on three tosses)

$=\dfrac16\times \dfrac56\times\dfrac56+\dfrac16\times\dfrac16\times\dfrac56+\dfrac16\times\dfrac16\times\dfrac16 \\=\dfrac{25}{216}+\dfrac{5}{216}+\dfrac{1}{216} \\=\dfrac{31}{216}$

From a deck of 52 cards:

a) A = {drawing of a picture card}; B ={drawing of a heart card} 

P(A) = 12/52 = 3/13, P(B) = 26/52 = 1/2, P(A$\cap$B) = 6/52 = 3/26

Now, P(A).P(B) = $\dfrac3{13}\times \dfrac12=\dfrac3{26}=P(A\cap B)$

Thus, they are independent.

b) If the king of hearts is missing, then:

P(A)=11/51, P(B)=25/51, P(A$\cap$B) = 5/51

Now, P(A).P(B) = $\dfrac{11}{51}\times \dfrac{25}{51}=\dfrac{275}{2601}\ne P(A\cap B)$

Thus, they are not independent.

c) What if a card, at random, is missing?

Case I: If missing card is a face card.

P(A)=11/51, P(B)=25/51, P(A$\cap$B) = 5/51

Now, P(A).P(B) = $\dfrac{11}{51}\times \dfrac{25}{51}=\dfrac{275}{2601}\ne P(A\cap B)$

Thus, they are not independent.

Case II: If missing card is not a face card but a heart card.

P(A)=12/51 = 4/17, P(B)=25/51, P(A$\cap$B) = 6/51 = 2/17

Now, P(A).P(B) = $\dfrac{4}{17}\times \dfrac{25}{51}=\dfrac{100}{867}\ne P(A\cap B)$

Thus, they are not independent.

Case III: If missing card is neither a face card nor a heart card.

P(A)=12/51 = 4/17, P(B)=26/51, P(A$\cap$B) = 6/51 = 2/17

Now, P(A).P(B) = $\dfrac{4}{17}\times \dfrac{26}{51}=\dfrac{104}{867}\ne P(A\cap B)$

Thus, they are not independent.

d) $P(getting\ the \ king\ of\ hearts)=\dfrac1{52}$