Answer to Question #265351 in Statistics and Probability for Honour

Question

Rolling a dice three times, evaluate the probability of having at least one 6. (5 marks)

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

1.

$$P(at \ least\ one\ 6)=1-(\dfrac{5}{6})^3=\dfrac{91}{216}$$

2.

a)

$$P(A)=\dfrac{12}{52}=\dfrac{3}{13}$$ $$P(B)=\dfrac{13}{52}=\dfrac{1}{4}$$ $$P(A\cap B)=\dfrac{3}{52}$$ $$P(A)P(B)=\dfrac{3}{13}(\dfrac{1}{4})=\dfrac{3}{52}=P(A\cap B)$$

The events $A$ and $B$ are statistically independent.

b)

$$P(A)=\dfrac{11}{51}$$ $$P(B)=\dfrac{12}{51}=\dfrac{4}{17}$$ $$P(A\cap B)=\dfrac{3}{51}=\dfrac{1}{17}$$ $$P(A)P(B)=\dfrac{11}{51}(\dfrac{4}{17})=\dfrac{44}{867}\not=\dfrac{1}{17}=P(A\cap B)$$

The events $A$ and $B$ are not statistically independent.

c)

i) The missing card is neither heart nor picture.

$$P(A)=\dfrac{12}{51}=\dfrac{4}{17}$$ $$P(B)=\dfrac{13}{51}$$ $$P(A\cap B)=\dfrac{4}{51}$$ $$P(A)P(B)=\dfrac{4}{17}(\dfrac{13}{51})=\dfrac{52}{867}\not=\dfrac{4}{51}=P(A\cap B)$$

The events $A$ and $B$ are not statistically independent.

ii) The missing card is picture but is not heart

$$P(A)=\dfrac{11}{51}$$ $$P(B)=\dfrac{13}{51}$$ $$P(A\cap B)=\dfrac{4}{51}$$ $$P(A)P(B)=\dfrac{11}{51}(\dfrac{13}{51})=\dfrac{143}{2601}\not=\dfrac{4}{51}=P(A\cap B)$$

The events $A$ and $B$ are not statistically independent.

iii) The missing card is heart but is not picture

$$P(A)=\dfrac{12}{51}=\dfrac{4}{17}$$ $$P(B)=\dfrac{12}{51}=\dfrac{4}{17}$$ $$P(A\cap B)=\dfrac{4}{51}$$ $$P(A)P(B)=\dfrac{4}{17}(\dfrac{4}{17})=\dfrac{16}{289}\not=\dfrac{4}{51}=P(A\cap B)$$

The events $A$ and $B$ are not statistically independent.

iv) The missing card is heart and is picture

$$P(A)=\dfrac{11}{51}$$ $$P(B)=\dfrac{12}{51}=\dfrac{4}{17}$$ $$P(A\cap B)=\dfrac{3}{51}=\dfrac{1}{17}$$ $$P(A)P(B)=\dfrac{11}{51}(\dfrac{4}{17})=\dfrac{44}{867}\not=\dfrac{1}{17}=P(A\cap B)$$

The events $A$ and $B$ are not statistically independent.

d)

$$P(King\ of\ hearts)=\dfrac{1}{52}$$