If 4-digit numbers greater than \(5,000\) are randomly formed from the digits \(0,1,3,5\), and \(7\), what is the probability of forming a number divisible by \(5 \) when (i)the digits are repeated? (ii)the repetition of digits is not allowed?
Solution and Explanation
(i) When the digits are repeated
Since four-digit numbers greater than \(5000\) are formed, the leftmost digit is either \(7\) or \(5\).
The remaining \(3\) places can be filled by any of the digits 0, 1, 3, 5, or 7 as repetition of digits is allowed.
∴Total number of 4-digit numbers greater than \(5000 = 2 × 5 × 5 × 5 - 1\)
\(= 250 - 1 = 249 \)
[In this case, \(5000 \)can not be counted; so \(1\) is subtracted]
A number is divisible by \(5\) if the digit at its unit place is either \(0\) or \(5\). ∴Total number of 4-digit numbers greater than 5000 that are divisible by 5 \(= 2 × 5 × 5 × 5 - 1 = 100 - 1 = 99\)
Thus, the probability of forming a number divisible by \(5\)5 when the digits are repeated is \(=\dfrac{99}{249}=\dfrac{33}{83}\)
(ii) When repetition of digits is not allowed
The thousands place can be filled with either of the two digits \(5\) or \(7\).
The remaining \(3\) places can be filled with any of the remaining 4 digits.
∴Total number of 4-digit numbers greater than \(5000 = 2 × 4 × 3 × 2 = 48 \)
When the digit at the thousands place is 5\(\), the units place can be filled only with 0 and the tens and hundreds places can be filled with any two of the remaining 3 digits.
∴Here, the number of 4-digit numbers starting with 5 and divisible by \(5 = 3 × 2 = 6\) When the digit at the thousands place is 7, the unit place can be filled in two ways (0 or \(5\)) and the tens and hundreds places can be filled with any two of the remaining 3 digits.
∴ Here, the number of 4-digit numbers starting with 7 and divisible by \(5 = 1 × 2 × 3 × 2 = 12\)5 \(=\) \(1 × 2 × 3 × 2 = 12 \)
∴Total number of 4-digit numbers greater than 5000 that are divisible by 5 = 6 + 12 = 18
Thus, the probability of forming a number divisible by 5 when the repetition of digits is not allowed is \(\dfrac{18}{48}=\dfrac{3}{8}\)
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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.