A bag contains 6 balls Two balls are drawn from it at random and both are found to be black The probability that the bag contains at least 5 black balls is
A bag contains 6 balls Two balls are drawn from it at random and both are found to be black The probability that the bag contains at least 5 black balls is
Show Hint
- $\frac{2}{7}$
- $\frac{3}{7}$
- $\frac{5}{7}$
- $\frac{5}{6}$
The Correct Option is C
Approach Solution - 1
The correct answer is (C) : \(\frac{5}{7}\)
\(\frac{^5C_2+^6C_2}{^2C_2+^3C_2+^4C_2+^5C_2+^6C_2}\)
\(=\frac{10+15}{1+3+6+10+15}=\frac{25}{35}\)
\(=\frac{5}{7}\)
Approach Solution -2
We are given a bag containing 6 balls, and two balls are drawn at random. The problem asks us to find the probability that the bag contains at least 5 black balls, given the probability that both balls drawn are black.
Let's define the events:
- The total number of ways to choose 2 balls from 6 is given by \( \binom{6}{2} \).
- The probability that both balls drawn are black depends on the number of black balls in the bag.
We need to calculate the number of ways to choose 2 black balls from the possible number of black balls. We are considering cases where there are 5 or 6 black balls in the bag.
The probability that both balls drawn are black is calculated for each of these cases. \[ \frac{\binom{5}{2} + \binom{6}{2}}{\binom{2}{2} + \binom{3}{2} + \binom{4}{2} + \binom{5}{2} + \binom{6}{2} + \binom{8}{2}} = \frac{10 + 15}{1 + 3 + 6 + 10 + 15} \] Now, simplifying the fraction: \[ = \frac{25}{35} = \frac{5}{7} \] Thus, the probability that the bag contains at least 5 black balls is \( \frac{5}{7} \). This approach involves calculating the total number of favorable outcomes (selecting 2 black balls) and dividing it by the total possible outcomes (selecting any 2 balls). We can also apply the principle of conditional probability to further understand how this result relates to the actual composition of the bag.
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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.