Solving Probability for Inequality \(x^2 + x > 50\)
The question asks us to find the probability that a natural number \(x\), chosen randomly from the first 100 natural numbers (i.e., \(1, 2, 3, \dots, 100\)), satisfies the inequality \(x^2 + x > 50\).
Understanding the Sample Space
The set of numbers from which we are choosing is the first 100 natural numbers. This means our sample space consists of the integers from 1 to 100.
- Total possible outcomes = 100 (since there are 100 numbers from 1 to 100).
Identifying Favorable Outcomes
We need to find the numbers \(x\) in the range \([1, 100]\) such that \(x^2 + x > 50\). Let's test values of \(x\) starting from 1:
- For \(x=1\): \(1^2 + 1 = 1 + 1 = 2\). Is \(2 > 50\)? No.
- For \(x=2\): \(2^2 + 2 = 4 + 2 = 6\). Is \(6 > 50\)? No.
- For \(x=3\): \(3^2 + 3 = 9 + 3 = 12\). Is \(12 > 50\)? No.
- For \(x=4\): \(4^2 + 4 = 16 + 4 = 20\). Is \(20 > 50\)? No.
- For \(x=5\): \(5^2 + 5 = 25 + 5 = 30\). Is \(30 > 50\)? No.
- For \(x=6\): \(6^2 + 6 = 36 + 6 = 42\). Is \(42 > 50\)? No.
- For \(x=7\): \(7^2 + 7 = 49 + 7 = 56\). Is \(56 > 50\)? Yes.
Since the function \(f(x) = x^2 + x\) increases as \(x\) increases for positive \(x\), all natural numbers \(x\) greater than or equal to 7 will satisfy the inequality \(x^2 + x > 50\).
So, the values of \(x\) from the first 100 natural numbers that satisfy the condition are \(7, 8, 9, \dots, 100\).
Counting the Favorable Outcomes
To count how many numbers are in the list \(7, 8, 9, \dots, 100\), we can use the formula: Number of terms = Last term - First term + 1.
- Number of favorable outcomes = \(100 - 7 + 1 = 94\).
Calculating the Probability
The probability of an event is calculated as:
\( \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
Plugging in our values:
\( \text{Probability} = \frac{94}{100} \)
Simplifying the Result
The fraction \(\frac{94}{100}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
\( \frac{94 \div 2}{100 \div 2} = \frac{47}{50} \)
Therefore, the probability that \(x^2 + x > 50\) when \(x\) is chosen randomly from the first 100 natural numbers is \(\frac{47}{50}\).
Similar Questions
- An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family in every year during these years, what is the average price of oil in rupees (approximately) per litre ?
- Let \(m = 77^n\). The index \(n\) is given a positive integral value at random. What is the probability that the value of \(m\) will have 1 in the units place ?
- What is \(P(A) + P(B) + P(C)\) equal to ?
- A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If \(3P(B) = 4P(A)\) and \(3P(C) = 2P(B)\), then what is \(P(A)\) equal to?
- A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?
- A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?
- The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?
- A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?
- Two perfect dice are thrown. What is the probability that the sum of the numbers on the faces is neither 9 nor 10?
- What is the probability that the number selected is divisible by 6 ?
Important Questions from Probability
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
