Multiple Probabilities
The probability of 2 things happening:
The probability of 2 independent things happening either one after the
other or together is the probability of the first thing happening
multiplied by the probability of the second thing
happening.
Example 1: Flipping a coin 2 times:
What is the probability of flipping a coin 2 times and it coming up
"heads" both times?
first second
toss: toss:
1 1 1
Answer: --- X --- = --- or 1 chance in 4 tries
2 2 4
Note: This answer of 1/4 is correct before you make the first toss.
After you toss the coin once, and it comes up heads, the probability
that
the second toss will also come up heads is 1/2 because the first toss is
in the past.
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Example 2: Drawing 2 cards at the same time
What is the probability of drawing 2 aces in a row out of a deck
of 52 cards,
if you don't put the first card back in the deck
before drawing the second card?
first second
draw: draw:
4 3 12 1
Answer: --- X --- = --- = --- or 1 chance in 221 tries
52 51 2652 221
Why is the probability of the second ace 3/51?
Well, after you draw the
first ace (you assume you got it) there are only 3 aces left and 51 cards.
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Example 3: Drawing 2 cards with replacement:
How would the probabilities change if you had put the first ace back in
the deck
before drawing the second time? Well, this makes the probability
of drawing an ace
on the second draw the same as on the first draw,
so the
probability of drawing 2 aces with replacement of the first ace
is:
first second
draw: draw:
4 4 16 1
Answer: --- X --- = --- = --- or 1 chance in 169 tries
52 52 2704 169
Since 1/169 is larger than 1/221, this means you are more likely to get 2
aces in a row
if you put your first ace back in the deck before you draw
your second card.
But you knew that!
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Now, here is a problem for you:
What is the probability of drawing an ace from a deck of cards
and a flipped coin turning up "heads"? = ______ X _____ = _____
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