An urn contains 8 black marbles and 4 white marbles. Two marbles are chosen at random and without replacement. Then the probability that both marbles are black is
\(7/33\)
\(2/3\)
\(7/11\)
\(14/33\)
\(21/143\)
The Correct Option is D
Solution and Explanation
Given data:
Black marble \(=8\)
White marbles \(=4\)
Then,
Step 1
Probability of drawing a black marble on the first draw = \dfrac{(Number of black marbles)}{(Total number of marbles)}
\(= \dfrac{8}{(8 + 4)} =\dfrac{ 2}{3}\)
Step 2
After the first black marble is drawn, there will be \(7\) black marbles left and \(4\) white marbles, making a total of \(11\) marbles for the second draw (since 1 marble has already been taken out).
Probability of drawing a black marble on the second draw, given that the first marble was black = (Number of remaining black marbles) / (Total number of remaining marbles)\( = 7 / 11\).
Step 3: Now the overall probability of drawing two black marbles
Probability of drawing two black marbles = Probability of first black marble × Probability of second black marble
\(= (2/3) × (7/11) =14/33\)
Hence the required answer is \(14/33\).
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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.