what is the probability of winning all the 6 digits in LOTTO 6/42
The correct answer and explanation is:
The probability of winning all 6 digits in a Lotto 6/42 game can be calculated using combinations from basic probability theory.
What is Lotto 6/42?
In Lotto 6/42, you select 6 distinct numbers out of a pool of 42 numbers (usually from 1 to 42). To win the jackpot, your chosen 6 numbers must match all the 6 numbers drawn by the lottery.
Calculating the probability:
- The total number of ways to choose 6 numbers out of 42 is given by the combination formula:
Total possible combinations=(426)=42!6!×(42−6)!\text{Total possible combinations} = \binom{42}{6} = \frac{42!}{6! \times (42-6)!}
- This number represents all possible different 6-number combinations you can pick.
- Since only one combination wins the jackpot, the probability of your chosen numbers matching exactly all 6 winning numbers is:
Probability of winning=1(426)\text{Probability of winning} = \frac{1}{\binom{42}{6}}
Let’s calculate:
(426)=42×41×40×39×38×376×5×4×3×2×1\binom{42}{6} = \frac{42 \times 41 \times 40 \times 39 \times 38 \times 37}{6 \times 5 \times 4 \times 3 \times 2 \times 1} =5,245,786,880720=7,059,052= \frac{5,245,786,880}{720} = 7,059,052
So, Probability of winning=17,059,052≈1.416×10−7\text{Probability of winning} = \frac{1}{7,059,052} \approx 1.416 \times 10^{-7}
Or about 1 in 7 million.
Explanation:
- The probability is extremely low because there are over 7 million possible unique combinations in the pool of 42 numbers.
- When you pick your 6 numbers, only one exact set matches the winning draw.
- The combination formula (nk)\binom{n}{k} is used here because order does not matter (picking 1,2,3,4,5,6 is the same as 6,5,4,3,2,1 in Lotto).
- This is why Lotto jackpots are so hard to win—your chance is only about 1 in 7 million per ticket.
- People often play multiple tickets or join lottery pools to increase their chances, but it remains very unlikely to hit the jackpot.