Answer
Verified
400.2k+ views
Hint:
Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes. Probability is the state of being probable and the extent to which something is likely to happen in the particular favourable situations. Here, we will find the probability of each of the three coins in the favourable pattern of Head, Head and Tail.
Complete step by step solution:
Give that: Coin A is the fair coin.
Therefore, the probability of occurring head for coin A is $P{(H)_A} = \dfrac{1}{2}$
(By definition – as out of the total two outcomes Head and Tail, our favourable outcomes Head will occur only once.)
When coin A is tossed, the probability of getting two heads and one tail is
$P{(HHT)_A} = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2}$
Therefore,
$P{(HHT)_A} = \dfrac{1}{8}\;{\text{ }}.......{\text{(1)}}$
Coin B is the biased coin and
The probability of occurring head for coin A is $P{(H)_B} = \dfrac{2}{3}$ (Given)
Therefore, the probability of getting tail is equal to one minus the probability of getting Head
$
P{(T)_B} = 1 - P(H){}_B \\
P{(T)_B} = 1 - \dfrac{2}{3} \\
P{(T)_B} = \dfrac{1}{3} \\
$
Now, by substituting the values -
When coin B is tossed, the probability of getting two heads and one tail is
$P{(HHT)_B} = \dfrac{2}{3} \times \dfrac{2}{3} \times \dfrac{1}{3}$
Therefore, $P{(HHT)_B} = \dfrac{4}{{27}}\,{\text{ }}.......{\text{(2)}}$
Similarly, for coin C
The probability of occurring head for coin C is
$P{(H)_C} = \dfrac{1}{3}$ (Given)
Therefore, the probability of getting tail is equal to one minus the probability of getting Head
$
P{(T)_B} = 1 - P(H){}_B \\
P{(T)_B} = 1 - \dfrac{1}{3} \\
P{(T)_B} = \dfrac{2}{3} \\
$
Now, by substituting the values -
When coin C is tossed, the probability of getting two heads and one tail is
$P{(HHT)_C} = \dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{2}{3}$
Therefore, $P{(HHT)_C} = \dfrac{2}{{27}}\;{\text{ }}........{\text{(3)}}$
From the Equations $(1),\,{\text{(2) \& (3)}}$
Now, combined probability of all the three coins is –
$
P{(HHT)_T} = P{(HHT)_A} + P{(HHT)_B} + P{(HHT)_C} \\
P{(HHT)_T} = \dfrac{1}{8} + \dfrac{4}{{27}} + \dfrac{2}{{27}} \\
$
Simplify by taking LCM
$
P{(HHT)_T} = \dfrac{{27 + 32 + 16}}{{216}} \\
P{(HHT)_T} = \dfrac{{75}}{{216}} \\
$
(Take from both the numerator and the denominator; also same number from numerator and denominator cancels each other)
$P{(HHT)_T} = \dfrac{{25}}{{72}}$
Now, the probability that the chosen coin was A, giving two heads and one tail when tossed is –
$
P(for{\text{ the coin A) = }}\dfrac{{P{{(HHT)}_A}}}{{P(HHT){}_T}} \\
P(for{\text{ the coin A)}} = \dfrac{{\dfrac{1}{8}}}{{\dfrac{{25}}{{72}}}} \\
$
Use the property, denominator or numerator goes in denominator and denominator’s denominator goes in numerator.
$\therefore P(for{\text{ the coin A) = }}\dfrac{{1 \times 72}}{{8 \times 25}}$
Now, simplify the right hand side of the equation –
$\therefore P(for{\text{ the coin A) = }}\dfrac{9}{{25}}$
Thus, the required answer is the probability that the chosen coin was A, giving two heads and one tail when tossed is –
$\therefore P(for{\text{ the coin A) = }}\dfrac{9}{{25}}$
Hence, from the given multiple choices, option A is the correct answer.
Note:
For this type of probability problems, just follow the general formula for probability and basic simplification properties for the fractions. Always remember the probability of any event lies between zero and one.
Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes. Probability is the state of being probable and the extent to which something is likely to happen in the particular favourable situations. Here, we will find the probability of each of the three coins in the favourable pattern of Head, Head and Tail.
Complete step by step solution:
Give that: Coin A is the fair coin.
Therefore, the probability of occurring head for coin A is $P{(H)_A} = \dfrac{1}{2}$
(By definition – as out of the total two outcomes Head and Tail, our favourable outcomes Head will occur only once.)
When coin A is tossed, the probability of getting two heads and one tail is
$P{(HHT)_A} = \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2}$
Therefore,
$P{(HHT)_A} = \dfrac{1}{8}\;{\text{ }}.......{\text{(1)}}$
Coin B is the biased coin and
The probability of occurring head for coin A is $P{(H)_B} = \dfrac{2}{3}$ (Given)
Therefore, the probability of getting tail is equal to one minus the probability of getting Head
$
P{(T)_B} = 1 - P(H){}_B \\
P{(T)_B} = 1 - \dfrac{2}{3} \\
P{(T)_B} = \dfrac{1}{3} \\
$
Now, by substituting the values -
When coin B is tossed, the probability of getting two heads and one tail is
$P{(HHT)_B} = \dfrac{2}{3} \times \dfrac{2}{3} \times \dfrac{1}{3}$
Therefore, $P{(HHT)_B} = \dfrac{4}{{27}}\,{\text{ }}.......{\text{(2)}}$
Similarly, for coin C
The probability of occurring head for coin C is
$P{(H)_C} = \dfrac{1}{3}$ (Given)
Therefore, the probability of getting tail is equal to one minus the probability of getting Head
$
P{(T)_B} = 1 - P(H){}_B \\
P{(T)_B} = 1 - \dfrac{1}{3} \\
P{(T)_B} = \dfrac{2}{3} \\
$
Now, by substituting the values -
When coin C is tossed, the probability of getting two heads and one tail is
$P{(HHT)_C} = \dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{2}{3}$
Therefore, $P{(HHT)_C} = \dfrac{2}{{27}}\;{\text{ }}........{\text{(3)}}$
From the Equations $(1),\,{\text{(2) \& (3)}}$
Now, combined probability of all the three coins is –
$
P{(HHT)_T} = P{(HHT)_A} + P{(HHT)_B} + P{(HHT)_C} \\
P{(HHT)_T} = \dfrac{1}{8} + \dfrac{4}{{27}} + \dfrac{2}{{27}} \\
$
Simplify by taking LCM
$
P{(HHT)_T} = \dfrac{{27 + 32 + 16}}{{216}} \\
P{(HHT)_T} = \dfrac{{75}}{{216}} \\
$
(Take from both the numerator and the denominator; also same number from numerator and denominator cancels each other)
$P{(HHT)_T} = \dfrac{{25}}{{72}}$
Now, the probability that the chosen coin was A, giving two heads and one tail when tossed is –
$
P(for{\text{ the coin A) = }}\dfrac{{P{{(HHT)}_A}}}{{P(HHT){}_T}} \\
P(for{\text{ the coin A)}} = \dfrac{{\dfrac{1}{8}}}{{\dfrac{{25}}{{72}}}} \\
$
Use the property, denominator or numerator goes in denominator and denominator’s denominator goes in numerator.
$\therefore P(for{\text{ the coin A) = }}\dfrac{{1 \times 72}}{{8 \times 25}}$
Now, simplify the right hand side of the equation –
$\therefore P(for{\text{ the coin A) = }}\dfrac{9}{{25}}$
Thus, the required answer is the probability that the chosen coin was A, giving two heads and one tail when tossed is –
$\therefore P(for{\text{ the coin A) = }}\dfrac{9}{{25}}$
Hence, from the given multiple choices, option A is the correct answer.
Note:
For this type of probability problems, just follow the general formula for probability and basic simplification properties for the fractions. Always remember the probability of any event lies between zero and one.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Net gain of ATP in glycolysis a 6 b 2 c 4 d 8 class 11 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE