Question

The probability of selecting green marble at random from a jar that contains only green, white, and yellow marbles is \[\dfrac{{\text{1}}}{{\text{4}}}\]
The probability of selecting white marble at random from the same jar is \[\dfrac{{\text{1}}}{3}\]. If the jar contains a total 10 yellow marbles,
Then what is the total number of marbles in the jar?
A. \[22\]
B. \[26\]
C. \[28\]
D. \[24\]

Answer 1

Hint: Firstly, calculate the probability of yellow marble selection. And as probability of any random event A is given as \[{\text{P(A) = }}\dfrac{{{\text{favourable outcomes}}}}{{{\text{total possible outcomes}}}}\].And hence by using this concept we can calculate the total number of marbles. Also the probability of the event ranges from \[{\text{0 < P(A) < 1}}\]. And if there are group of events than probability sum of all that event is one.

Complete step by step answer:

Calculating the probability of all the individual and then the summation of all will be equal to one. Using these two concepts we can approach the given.
Let the probability of selection of green , white and yellow marbles be \[{\text{P(G),P(W),P(Y)}}\]
Sum of all the events of the probability is \[{\text{P(G) + P(W) + P(Y) = 1}}\]
So, putting the value of probabilities of given marbles we get,
\[\dfrac{1}{4}{\text{ + }}\dfrac{1}{3}{\text{ + P(Y) = 1}}\]
\[\begin{gathered}
  {\text{P(Y) = 1 - }}\dfrac{{\text{1}}}{{\text{4}}}{\text{ - }}\dfrac{{\text{1}}}{{\text{3}}} \\
  {\text{P(Y) = }}\dfrac{{\text{5}}}{{{\text{12}}}} \\
\end{gathered} \]
Now, \[{\text{P(Y) = }}\dfrac{{{\text{favourable outcomes}}}}{{{\text{total possible outcomes}}}}{\text{ = }}\dfrac{{\text{5}}}{{{\text{12}}}}\]
Now, as we know that there are \[10\]marbles of yellow colour which is favourable event for selection of yellow marble.
And so the total number of marbles will be
\[\begin{gathered}
  {\text{P(Y) = }}\dfrac{{{\text{10}}}}{{{\text{total possible outcomes}}}}{\text{ = }}\dfrac{{\text{5}}}{{{\text{12}}}} \\
  \Rightarrow {\text{n(total possible outcomes) = 24}} \\
\end{gathered} \]
Hence, option (d) is our correct answer.

Note: Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. Applying the step where the total probability of even is one carefully. Probability is the ratio of outcomes to the total. So, use it carefully.