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.
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\n\\\\=\\dfrac{25}{216}+\\dfrac{5}{216}+\\dfrac{1}{216}\n\\\\=\\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}"
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