A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If denotes the number of tosses of the coin, then the mean of is [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

Q.
A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is [2023]

1 $\frac{21}{16}$

2 $\frac{37}{16}$

3 $\frac{81}{64}$

4 $\frac{15}{16}$

Ans.
(1)

$Let x be the probability of getting tail.$

$\Rightarrow P (T) = x$ $∴$ $P (H) = 3 x .$

$We know that sum of all probabilities of events of an experiment = 1$

$∴$ $x + 3 x = 1 \Rightarrow x = \frac{1}{4}$ $\Rightarrow P (T) = \frac{1}{4}, P (H) = \frac{3}{4}$

X 1 2 3

P(X) $\frac{3}{4}$ $\frac{1}{4} \times \frac{3}{4}$ ${(\frac{1}{4})}^{3} + {(\frac{1}{4})}^{2} \times (\frac{3}{4})$

$Now, Mean \bar{X} = \sum x_{i} P (x_{i})$ $= 1 \times \frac{3}{4} + 2 \times \frac{1}{4} \times \frac{3}{4} + 3 (\frac{1}{64} + \frac{3}{64})$

$= \frac{3}{4} + \frac{6}{16} + \frac{3}{16} = \frac{21}{16}$

Solution

Q.
A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is [2023]

1 $\frac{21}{16}$

2 $\frac{37}{16}$

3 $\frac{81}{64}$

4 $\frac{15}{16}$

Want Us to Email you About Special Offers & Updates?

X	1	2	3
P(X)	$\frac{3}{4}$	$\frac{1}{4} \times \frac{3}{4}$	${(\frac{1}{4})}^{3} + {(\frac{1}{4})}^{2} \times (\frac{3}{4})$

Solution

Q. A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is [2023]

1 2116

2 3716

3 8164

4 1516

Ans. (1) Let x be the probability of getting tail. ⇒P(T)=x ∴ P(H)=3x. We know that sum of all probabilities of events of an experiment=1 ∴ x+3x=1⇒x=14⇒P(T)=14,P(H)=34 X 1 2 3 P(X) 34 14×34 (14)3+(14)2×(34) Now, Mean X¯=∑xiP(xi)=1×34+2×14×34+3(164+364) =34+616+316=2116

Want Us to Email you About Special Offers & Updates?

Q.
A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is [2023]

1 $\frac{21}{16}$

2 $\frac{37}{16}$

3 $\frac{81}{64}$

4 $\frac{15}{16}$