If f(x) = 3x - 2 and g(x) = 2x + 1, find (f+ g)(x).
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Answer:
The selection of the two marbles are <u>dependent</u> because the selection of the pink marble <u>changes</u> the probability that a green marble is drawn next.
If the pink marble had been put back into the bag, the selection of the two marbles would have been <u>independent</u>
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Explanation:
The key phrase to focus on here is "This marble is placed on a table". It implies that we do not put it back into the bag, and also implies we don't put in a replacement either. Hence we consider this scenario as "no replacement".
In no replacement situations, we always have dependent events. The second selection depends on the first one.
The probability of picking green is 6/11 because there are 6 green out of 5+6 = 11 total. But if we picked a pink one first and didn't put it back, then the 11 drops to 10 and we get 6/10 = 3/5 as the new probability.
In other words,
- before the pink marble is selected, the chances of green are 6/11
- after the pink marble is selected, and not put back, the chances of green are now 6/10 = 3/5
Because this probability changes, it's sufficient evidence to say the event of picking green on the second selection is dependent on the first selection of selecting pink.
If the marble had been put back, then we'd revert back to the original state of the bag. That would mean the chances of green would stay at 6/11. Therefore, situations with replacement lead to independent events.
