A coin is tossed three times in succession. If `E` is the event that there are at least two heads and `F` is the event in which first throw is a head, then find `P(E//F)dot`
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A coin is tossed three times, consider the following events. A : No head appears, B: Exactly one head appears and C: Atleast two appear. Do they form
The sample space of the experiment is `S = {HHH,HHT,HTH,THH,HT T,THT,T TH,T T T}` `A = {T T T}` `B = {HT T, THT, T TH}` `C = {HHT, HTH,...
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A coin is tossed three times, where (i) A : head on third toss,B: heads on first two tosses (ii) A: at least two heads, B : at most two heads (iii) A
If a coin is tossed three times, then the sample S is `S={HHH,HHT, HTH, HT T ,THH,THT, T TH, T T T }` `thereforen(S)=8.` (i) `A={HHH, HTH, THH, T TH]`...
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An unbiased normal coin is tossed `n` times. Let `E_1:` event that both heads and tails are present `n` tosses. `E_2:` event that the coin shows up he
`P(E_(1))=1-["P(all heads)+P(all tails")]` `=1-[(1)/(2^(n))+(1)/(2^(n))]=1-(1)/(2^(n-1))` `P(E_(2))=P("no head)+P(exactly one head")` `=(1)/(2^(n))+^(""n)C_(1).(1)/(2^(n))=(n)/(2^(n))` `P(E_(1)nnE_(2))=("exactly one head and"(n-1)till)` `=^(""n)C_(1).(1)/(2).(1)/(2^(n-1))=(n)/(2^(n))` If `E_(1)and E_(2)` are indipendent, then `P(E_(1)nnE_(2))=P(E_(1)).P(E_(2))` `implies(n)/(2^(n))=(1-(1)/(2^(n-1)))((n+1)/(2^(n)))` `impliesn=(1-(1)/(2^(n-1)))(n+1)`
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A coin is tossed three times. Event A: two heads appear Event B: last should be head Then identify whether events A and B are independent or not.
Correct Answer - Not independent `P(A)=3/8and P(B)=1/2` `P(A)P(B) =3/8xx1/2=3/16` `P(Annb)=2/8=1/4neP(A)P(B)` Hence, A and B aer not independent.
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A fair coin is tossed `n` times. if the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then find the value
Correct Answer - `n=14` According to question `""^(n)C_(6)((1)/(2))^(6)((1)/(2))^(n-6)=""^(n)C_(8)((1)/(2))^(8)((1)/(2))^(n-8)` `or ""^(n)C_(6)((1)/(2))^(n)=""^(n)C_(8)((1)/(2))^(n)` `or""^(n)C_(6)=""^(n)C_(8)=""^(n)C_(n-8)` `implies6=n-8orn=14`
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A coin is tossed `2n` times. The chance that the number of times one gets head is not equal to the number of times one gets tails is `((2n !))/((n !)^
Correct Answer - C The required probability is 1- probability of getting equal number of heads and tails `=1-""^(2n)C_(n)((1)/(2))^(n)((1)/(2))^(2n-n)` `=1-((2n)!)/((n!)^(2))xx(1)/(4^(n))`
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A fair coin is flipped n times. Let E be the event "a head is obtained on the first flip" and let `F_(k)` be the event "exactly k heads are obtained".
Correct Answer - 2 Here `P(E)=1/2and P(F_(k))=""^(n)C_(k).(1)/(2^(n))` Also, `P(EnnF_(k))=p` (exactly k heads are obtained and head obtained in first filp) `=1/2""^(n-1)C_(k-1)((1)/(2))^(n-1)` Events E and `f_(k)` are independent. Therefore, `P(EnnF_(k))=P(E).P(F_(k))` `or ""^(n-1)C_(k-1)xx(1)/(2^(n))=1/2xx""^(n)C_(k)(1)/(2^(n))`...
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An unbiased normal coin is tossed n times. Let `E_(1):` event that both heads and tails are present in n tosses. `E_(2):` event that the coin shows up
Correct Answer - 3 `P(E_(1))=1-[P ("all heads)=P(all tails")]` `=1-[(1)/(2^(n))+(1)/(2^(n))]=1-(1)/(2^(n-1))` `P(E_(2))=["P(no head)+P (exactly one head)"]` `=(1)/(2^(n))+""^(n)C_(1)xx(1)/(2^(n-1))=(n)/(2^(n))` If `E_(1) and E_(2)` are independent, then `(n)/(2^(n))(1-(1)/(2^(n-1)))((n+1)/(2^(n)))` `orn =(1-(1)/(2^(n-1)))(n+1)` `or n=n+1-(n+1)/(2^(n-1))` `or n+1=2^(n-1)` `or n=3`
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A fair coin is tossed `n` times. Let `a_(n)` denotes the number of cases in which no two heads occur consecutively. Then which of the following is not
Correct Answer - C `(c )` The cases for `a_(1){H,T}` i.e, `a_(1)=2` The cases for `a_(2){HT,TH,TT}`, `a_(2)=3` For `n ge 3` , if the first outcome is `H`, then next just...
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Three coins are tossed 100 times, and three heads one head occurred 14 times and head did not occur 23 times. Find the probability of getting more tha
Correct Answer - `(63)/(100)`
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