Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event

Question

Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are
(i) Mutually exclusive? (ii) Simple? (iii) Compound?

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Delilah 1 month 2021-08-16T14:34:48+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-16T14:36:27+00:00

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    Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

    (i) Mutually exclusive?

    (ii) Simple?

    (iii) Compound?

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    ➡️Since either coin can turn up Head (H) or Tail (T), are the possible outcomes.

    ➡️But, now three coins are tossed once so the possible sample space contains,

    ➡️S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTH}

    ➡️Now,

    ➡️A: ‘three heads’

    ➡️A= (HHH)

    ➡️B: “two heads and one tail”

    ➡️B= (HHT, THH, HTH)

    ➡️C: ‘three tails’

    ➡️C= (TTT)

    ➡️D: a head shows on the first coin

    ➡️D= (HHH, HHT, HTH, HTT)

    ➡️(i) Mutually exclusive

    ➡️A ∩ B = (HHH) ∩ (HHT, THH, HTH)

    ➡️= φ

    ➡️Therefore, A and C are mutually exclusive.

    ➡️A ∩ C = (HHH) ∩ (TTT)

    ➡️= φ

    ➡️There, A and C are mutually exclusive.

    ➡️A ∩ D = (HHH) ∩ (HHH, HHT, HTH, HTT)

    ➡️= (HHH)

    ➡️A ∩ D ≠ φ

    ➡️So they are not mutually exclusive

    ➡️B ∩ C = (HHT, HTH, THH) ∩ (TTT)

    ➡️= φ

    ➡️Since there is no common element in B & C, so they are mutually exclusive.

    ➡️B ∩ D = (HHT, THH, HTH) ∩ (HHH, HHT, HTH, HTT)

    ➡️= (HHT, HTH)

    ➡️B ∩ D ≠ φ

    ➡️Since there are common elements in B & D,

    ➡️So, they not mutually exclusive.

    ➡️C ∩ D = (TTT) ∩ (HHH, HHT, HTH, HTT)

    ➡️= φ

    ➡️Since there is no common element in C & D,

    ➡️So they are not mutually exclusive.

    ➡️(ii) Simple event

    ➡️If an event has only one sample point of a sample space, it is called a simple (or elementary) event.

    ➡️A = (HHH)

    ➡️C = (TTT)

    ➡️Both A & C have only one element,

    ➡️so they are simple events.

    ➡️(iii) Compound events

    ➡️If an event has more than one sample point, it is called a Compound event

    ➡️B= (HHT, HTH, THH)

    ➡️D= (HHH, HHT, HTH, HTT)

    ➡️Both B & D have more than one element,

    ➡️So, they are compound events.

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    0
    2021-08-16T14:36:38+00:00

    Answer:

    ➡️Since either coin can turn up Head (H) or Tail (T), are the possible outcomes.

    ➡️But, now three coins are tossed once so the possible sample space contains,

    ➡️S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTH}

    ➡️Now,

    ➡️A: ‘three heads’

    ➡️A= (HHH)

    ➡️B: “two heads and one tail”

    ➡️B= (HHT, THH, HTH)

    ➡️C: ‘three tails’

    ➡️C= (TTT)

    ➡️D: a head shows on the first coin

    ➡️D= (HHH, HHT, HTH, HTT)

    ✴➡️(i) Mutually exclusive

    ➡️A ∩ B = (HHH) ∩ (HHT, THH, HTH)

    ➡️= φ✔

    ➡️Therefore, A and C are mutually exclusive.

    ➡️A ∩ C = (HHH) ∩ (TTT)

    ➡️= φ✔

    ➡️There, A and C are mutually exclusive.

    ➡️A ∩ D = (HHH) ∩ (HHH, HHT, HTH, HTT)

    ➡️= (HHH)✔

    ➡️A ∩ D ≠ φ

    ➡️So they are not mutually exclusive

    ➡️B ∩ C = (HHT, HTH, THH) ∩ (TTT)

    ➡️= φ✔

    ➡️Since there is no common element in B & C, so they are mutually exclusive.

    ➡️B ∩ D = (HHT, THH, HTH) ∩ (HHH, HHT, HTH, HTT)

    ➡️= (HHT, HTH)✔

    ➡️B ∩ D ≠ φ

    ➡️Since there are common elements in B & D,

    ➡️So, they not mutually exclusive.

    ➡️C ∩ D = (TTT) ∩ (HHH, HHT, HTH, HTT)

    ➡️= φ✔

    ➡️Since there is no common element in C & D,

    ➡️So they are not mutually exclusive.

    ✴➡️(ii) Simple event

    ➡️If an event has only one sample point of a sample space, it is called a simple (or elementary) event.

    ➡️A = (HHH)✔

    ➡️C = (TTT)✔

    ➡️Both A & C have only one element,

    ➡️so they are simple events.

    ✴➡️(iii) Compound events

    ➡️If an event has more than one sample point, it is called a Compound event

    ➡️B= (HHT, HTH, THH)✔

    ➡️D= (HHH, HHT, HTH, HTT)✔

    ➡️Both B & D have more than one element,

    ➡️So, they are compound events.

    Step-by-step explanation:

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