A bag contains six balls of different colours.Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$ .If $p: q=m: n$, where $m$ and $n$ are coprime, then $m+n$ is equal to______
Correct Answer: 14
Approach Solution - 1
So , the correct answer is 14.
Approach Solution -2
Calculate p
The probability that both balls drawn are of the same colour:
1. There are 6 different colours.
2. The probability of drawing a ball of any specific colour in the first draw is \(\frac{1}{6}\).
3. Since we replace the ball, the probability of drawing the same colour again is also \(\frac{1}{6}\).
Therefore, the probability p of drawing two balls of the same colour is:
\(p = 6 \times \left(\frac{1}{6} \times \frac{1}{6}\right) = 6 \times \frac{1}{36} = \frac{6}{36} = \frac{1}{6}\)
Calculate q
The probability that exactly three out of four balls drawn are of the same color:
1. Choose one colour out of 6 for the three balls: 6 ways.
2. Choose the position for the different ball: \(\binom{4}{1} = 4\) ways.
3. Choose a different colour for this ball: 5 ways (since there are 5 remaining colors).
The number of favorable outcomes is:
\(6 \times 4 \times 5 = 120\)
The total number of possible outcomes when drawing four balls with replacement is:
\(6^4 = 1296\)
Thus, the probability q is:
\(q = \frac{120}{1296} = \frac{10}{108} = \frac{5}{54}\)
Ratio p:q
Given \(p = \frac{1}{6}\) and \(q = \frac{5}{54}\), we find the ratio \(\frac{p}{q}\):
\(\frac{p}{q} = \frac{\frac{1}{6}}{\frac{5}{54}} = \frac{1}{6} \times \frac{54}{5} = \frac{54}{30} = \frac{9}{5}\)
Therefore, the ratio \(p : q = 9 : 5\).
Since m = 9 and n = 5, and they are coprime, the sum m + n is:
\(m + n = 9 + 5 = 14\)
So, the answer is 14.
Top Questions on Probability
- A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
Let \( A_1 \): People with good health,
\( A_2 \): People with average health,
and \( A_3 \): People with poor health.
During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.
Based upon the above information, answer the following questions:
(i) A person was tested randomly. What is the probability that he/she has contracted the disease?}
(ii) Given that the person has not contracted the disease, what is the probability that the person is from category \( A_2 \)?- If A and B are two events such that $P(B) = \frac{1}{5}$, $P(A | B) = \frac{2}{3}$ and $P(A \cup B) = \frac{3}{5}$, then $P(A)$ is :
- If \( P(A) = \frac{1}{5}, P(B) = \frac{3}{5} \) and \( P(A \cap B) = \frac{2}{5} \), then \( P(A' \cup B') \) is :
- Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly from Bag I is transferred to Bag II and then a ball is drawn randomly from Bag II. Find the probability that the ball drawn is white.
Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.