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Chapter 33: Probability – Exercise 33.3 Probability – Exercise 33.3 – Q.1 (i) It is valid as each p(w1) lies between 0 to 1 and sum of p(w1) = 1 (ii) It is valid as each p(w1) lies between 0 to 1 and sum of p(w1) = 1 (iii) It is not valid as sum of p(w1) = 2.8 ≠1 (iv) It is not valid as p(w7) = 15/14>1 Which is impossible (i), (ii) Probability – Exercise 33.3 – Q.2 (i) ∵ a die is thrown Let E be the event of getting prime number ∴ E = {2, 3, 5} n(E) = 3 (ii) E = {2, 4} ∴ n(E) = 2 (iii) E = {2, 4, 6, 3} ⟹ n(E) = 4 Probability – Exercise 33.3 – Q.3 Since a pair of dice have been thrown ∴ Number of elementary events in sample space is 62 = 36 (i) Let E be the event that the sum 8 appear on the faces of dice ∴ E = {(2, 6), (3, 5), (4, 9), (5, 3), (6, 2)} ∴ n(E) = 5 (ii) a doublet Let E be the event that a doublet appears on the faces of dice ∴ E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} ⟹ n(E) = 6 (iii) a doublet of prime numbers Let E be the event that a doublet of prime number appear. ∴E = {(2, 2), (3, 3), (5, 5)} n(E) = 3 (iv) a doublet of odd numbers Let E be the event that a doublet of odd numbers appear. ∴E = {(1, 1), (3, 3), (5, 5)} ⟹ n(E) = 3 (v) a sum greater than 9 Let E be the event that a sum greater than appear ∴E = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)} ∴ n(E) = 6 (vi) an even number on first Let E be the event that an even number on the first dice appear Which means any number can be appear on second dice, ∴ n(E) = 18 (vii) an even number on one and a multiple of 3 on the other. Let E be the event that an even number on one and multiple of 3 on the other appears. ∴ E = {(2,3), (2, 6), (4, 3), (4, 6), (6, 3), (6, 6), (3, 2), (3, 4) (3, 6), (6, 2), (6, 4)} ∴ n(E) = 11 be the event that either 9 or 11 as the sum of number appear on the faces of dice. = {(3, 6), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5)} (ix) a sum less than 6. Let E be the event that less than 6 as a sum offer on the faces of dice. ∴ E= {(1,1), (1, 2), (1, 3), (1, 4), (2,1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)} ∴ n(E) = 10 (x) a sum less than 7. Let E be the event that less than 7 as a sum appears on the faces of dice. n(E) = 15 (xi) a sum more than 7. Let E be the event that a sum more than 7 appear on the faces of dice. ⟹ n(E) = 15 (xii) neither a doublet nor a total of 10. Let E be the event that neither a doublet nor a sum of 10 appear on the faces of dice. be the event that either a doublet or a sum of 10 appear on the faces of dice. {(1, 1), (2, 2), (3, 3), (4, 6), (5, 5), (6, 4), (6, 6)} (xiii) odd number on the first and 6 on the second. Let E be the event that an odd number on the first and 6 on the second appear on the faces of dice. ∴ E = {(1, 6), (3, 6), (5, 6)} n(E) = 3 (xiv) a number greater than 4 on each die. Let E be the event that a number greater than 4 appear on each dice ∴ E = {(5, 5), (5, 6), (6, 5), (6, 6)} ⟹ n(E) = 4 (xv) a total of 9 or 11. Let E be the event that a total of 9 or 11 appear on faces of dice. ∴ E = {(3, 6), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5)} ⟹ n(E) = 6 (xvi) a total greater than 8. Let E be the event that sum greater than 8 appear. ∴ E = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)} ∴ n(E) = 10 Probability – Exercise 33.3 – Q.4 ∵ Three dice are thrown ∴ n(S) = 63 = 216 Let E be the event of getting total of if 17 or 18 ∴ E = {(6, 6, 5), (6, 5, 6), (5, 6, 6), (6, 6, 6)} ⟹ n(E) = 4 Probability – Exercise 33.3 – Q.5 Three coins are tossed ∴ n(S) = 23 = 8 (i) E be the event of getting exactly two heads ∴ E = {HHT, HTH, THH} ∴ n (E) = 3 (ii) E at least two heads (two or 3 heads) ∴ E = {HHH, HHT, THH, HTH} n(E) = 4 (iii) at least one head and one tail ∴ E = {HTT, THT, TTH, HHT, HTH, THH} ∴ n(E) = 6
(i) It is valid as each p(w1) lies between 0 to 1 and sum of p(w1) = 1
(ii) It is valid as each p(w1) lies between 0 to 1 and sum of p(w1) = 1
(iii) It is not valid as sum of p(w1) = 2.8 ≠1
(iv) It is not valid as p(w7) = 15/14>1
Which is impossible
(i), (ii)
(i) ∵ a die is thrown
Let E be the event of getting prime number
∴ E = {2, 3, 5}
n(E) = 3
(ii) E = {2, 4} ∴ n(E) = 2
(iii) E = {2, 4, 6, 3}
⟹ n(E) = 4
Since a pair of dice have been thrown
∴ Number of elementary events in sample space is 62 = 36
(i) Let E be the event that the sum 8 appear on the faces of dice
∴ E = {(2, 6), (3, 5), (4, 9), (5, 3), (6, 2)}
∴ n(E) = 5
(ii) a doublet
Let E be the event that a doublet appears on the faces of dice
∴ E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
⟹ n(E) = 6
(iii) a doublet of prime numbers
Let E be the event that a doublet of prime number appear.
∴E = {(2, 2), (3, 3), (5, 5)}
(iv) a doublet of odd numbers
Let E be the event that a doublet of odd numbers appear.
∴E = {(1, 1), (3, 3), (5, 5)}
⟹ n(E) = 3
(v) a sum greater than 9
Let E be the event that a sum greater than appear
∴E = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}
∴ n(E) = 6
(vi) an even number on first
Let E be the event that an even number on the first dice appear
Which means any number can be appear on second dice,
∴ n(E) = 18
(vii) an even number on one and a multiple of 3 on the other.
Let E be the event that an even number on one and multiple of 3 on the other appears.
∴ E = {(2,3), (2, 6), (4, 3), (4, 6), (6, 3), (6, 6), (3, 2), (3, 4) (3, 6), (6, 2), (6, 4)}
∴ n(E) = 11
be the event that either 9 or 11 as the sum of number appear on the faces of dice.
= {(3, 6), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5)}
(ix) a sum less than 6.
Let E be the event that less than 6 as a sum offer on the faces of dice.
∴ E= {(1,1), (1, 2), (1, 3), (1, 4), (2,1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
∴ n(E) = 10
(x) a sum less than 7.
Let E be the event that less than 7 as a sum appears on the faces of dice.
n(E) = 15
(xi) a sum more than 7.
Let E be the event that a sum more than 7 appear on the faces of dice.
⟹ n(E) = 15
(xii) neither a doublet nor a total of 10.
Let E be the event that neither a doublet nor a sum of 10 appear on the faces of dice.
be the event that either a doublet or a sum of 10 appear on the faces of dice.
{(1, 1), (2, 2), (3, 3), (4, 6), (5, 5), (6, 4), (6, 6)}
(xiii) odd number on the first and 6 on the second.
Let E be the event that an odd number on the first and 6 on the second appear on the faces of dice.
∴ E = {(1, 6), (3, 6), (5, 6)}
(xiv) a number greater than 4 on each die.
Let E be the event that a number greater than 4 appear on each dice
∴ E = {(5, 5), (5, 6), (6, 5), (6, 6)}
(xv) a total of 9 or 11.
Let E be the event that a total of 9 or 11 appear on faces of dice.
∴ E = {(3, 6), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5)}
(xvi) a total greater than 8.
Let E be the event that sum greater than 8 appear.
∴ E = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}
∵ Three dice are thrown
∴ n(S) = 63 = 216
Let E be the event of getting total of if 17 or 18
∴ E = {(6, 6, 5), (6, 5, 6), (5, 6, 6), (6, 6, 6)}
Three coins are tossed
∴ n(S) = 23 = 8
(i) E be the event of getting exactly two heads
∴ E = {HHT, HTH, THH}
∴ n (E) = 3
(ii) E at least two heads (two or 3 heads)
∴ E = {HHH, HHT, THH, HTH}
n(E) = 4
(iii) at least one head and one tail
∴ E = {HTT, THT, TTH, HHT, HTH, THH}
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Chapter 33: Probability – Exercise 33.1...
Chapter 33: Probability – Exercise 33.2...
Chapter 33: Probability – Exercise 33.4...