Question

If \(S=\{1,2,....,50\}\), two numbers \(\alpha\) and \(\beta\) are selected at random find the probability that product is divisible by 3 :

• A. \(\frac{664}{1225}\)
• B. \(\frac{646}{1225}\)
• C. \(\frac{527}{1225}\)
• D. \(\frac{461}{1225}\)

Hint:

For probability problems involving conditions like "at least one" or "divisible by," it is often much simpler to calculate the probability of the complementary event ("none" or "not divisible by") and subtract it from 1.

Answer 1

The Correct Option is A

Step 1: Understanding the Question: 
We are selecting two distinct numbers from the set S = \(\{1, 2, ..., 50\}\). We need to find the probability that their product, \(\alpha\beta\), is a multiple of 3. It's often easier to calculate the probability of the complementary event. 

Step 2: Complementary Event: 
The complementary event is that the product \(\alpha\beta\) is NOT divisible by 3. This occurs if and only if neither \(\alpha\) nor \(\beta\) is divisible by 3. 

Step 3: Total Number of Outcomes: 
The total number of ways to choose two distinct numbers from 50 is given by the combination formula: \[ \text{Total Outcomes} = ^{50}C_2 = \frac{50 \times 49}{2 \times 1} = 25 \times 49 = 1225 \] 
Step 4: Favorable Outcomes for the Complementary Event: 
First, we count the numbers in S that are not divisible by 3. 
Numbers divisible by 3 in S are \(\{3, 6, 9, ..., 48\}\). The number of such terms is \(\frac{48}{3} = 16\). 
Numbers NOT divisible by 3 in S are \(50 - 16 = 34\). For the product \(\alpha\beta\) to not be divisible by 3, both \(\alpha\) and \(\beta\) must be chosen from these 34 numbers. The number of ways to choose 2 numbers from these 34 numbers is: \[ \text{Favorable Outcomes for Complement} = ^{34}C_2 = \frac{34 \times 33}{2 \times 1} = 17 \times 33 = 561 \] 
Step 5: Calculating Probabilities: 
The probability of the complementary event (product not divisible by 3) is: \[ P(\text{not divisible by 3}) = \frac{\text{Favorable Outcomes for Complement}}{\text{Total Outcomes}} = \frac{561}{1225} \] The probability of the desired event (product is divisible by 3) is 1 minus the probability of the complementary event: \[ P(\text{divisible by 3}) = 1 - P(\text{not divisible by 3}) = 1 - \frac{561}{1225} \] \[ P(\text{divisible by 3}) = \frac{1225 - 561}{1225} = \frac{664}{1225} \] 
Step 6: Final Answer: 
The probability that the product is divisible by 3 is \(\frac{664}{1225}\).