Cheryn
if x can solve with probability of 25 % and Y can at 50 % then combined together how come it is lesser than both????????? ie here 1/8.. i cannot feel this physically . could somebody explain this??
Cheryn, maybe what follows will help, because intuition is important: Think about how strictly or restrictively "success" is defined. (Success = desired outcome = win = passing the test.)
The probability of "success" here is restrictive because BOTH have to pass. If only X
OR Y had to pass, success would be easier.
To win, "BOTH this AND that must happen." That is a stricter definition of success than "to win, EITHER this OR that must happen."
In the first case,
two people must succeed. In the second case (OR),
only one person OR the other person must succeed. It's harder to get two wins than it is to get one win.
Look at the difference: if the question were "What is the probability of X
or Y passing the test?" The answer:
\(\frac{1}{2} + \frac{1}{4}=\frac{3}{4}\)
One OR the other must pass? Easier, less restrictive than "both must pass."
Different scenario, but it works exactly the same way.
A coin toss. What is the probability that one coin, flipped twice, will land on tails both times? Success = tails on the first flip AND on the second flip
P (tails) on the first flip is \(\frac{1}{2}\)
P (tails) on second flip = \(\frac{1}{2}\)
Events are independent.
Multiply:\(\frac{1}{2}* \frac{1}{2}=\frac{1}{4}\)
The probability of having both flips come up tails, \(\frac{1}{4}\), is lower (smaller) than the probability of having just one flip come up tails (\(\frac{1}{2}\)).
Again, that is because success is defined more restrictively. It is harder to get two tails on two flips than it is to get one tail on two flips; you have to "beat the odds" twice, not once. Lower probability.
Most probabilities are fractions between 0 and 1. When those fractions are multiplied, they get smaller. That fits.
It can be a little counterintuitive if you focus on AND. "And" might seem as if it should produce a better chance of success than "or." Maybe focus instead on: the definition of success, and how success is achieved.
Almost always, (BOTH must win) will be harder (lower probability) than (ONE OR THE OTHER must win).
Hope that helps.