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Topic: Mutually Exclusive and Complementary Events


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Subject:   RE: Mutually Exclusive and Complementary Events
Author: Craig
Date: Oct 27 2004
In looking over the standards (I pasted in from your original message), I think
a quick-brush sketch of independence in all that is necessary.  I agree with
"mathman"'s colleague that independence can be a very cerebral topic.  My
juniors and seniors in AP Statistics struggle with it... I can't imagine
expecting middle schoolers to master independence.  It appears from the
standards that "outcome of one trial does not affect outcome of another trial"
is as precise a definition (and understanding) of independence as your students
need.

I don't particularly like your example with "complementary" vs. "independent."
It adds layers of confusion; I would go the other way, and make sure that with
"compl." we're talking about different outcomes from the same experiment, while
with independent we're (usually) talking of the outcomes of different
experiments.  While getting a head and getting a tail are complementary events
with regards to the outcomes of a single coin toss, it doesn't make sense to
talk about a single outcome (either heads or tails) as being independent...
independent from what?  Getting a head on one coin toss is entirely independent
of getting a tail on another coin toss, though.  Said another (more complicated)
way:  "mutually exclusive" and "complementary" refer to the sample space;
"independent" refers to the trial, or the selection of an item from the sample
space.  

For example, if Mom has a cookie jar with 3 burnt cookies and 5 good cookies,
knowing that Susy got a good cookie affects the outcome of whether I get a good
cookie if I'm stuck getting my cookie after Susy.  My getting a good cookie and
Susy getting a good cookie are dependent events.  On the other hand, if we're
playing Monopoly and I roll doubles on my turn, you are no more or no less
likely to roll doubles on your turn.  Rolls of the dice are independent.

I'm not sure from reading the standards whether you need students to be able to
figure out the probability of getting two heads if you toss a coin twice... the
"multiplication counting principle" uses the fact that coin tosses are
independent to get (1/2)*(1/2) = 1/4.  On the other hand, the probability of
getting two burnt cookies also uses the multiplication counting principle: it's
(3/8)*(2/7) = 5/56.  Note that 2/7 is NOT equal to 3/8... the probability for
the second cookie depends on the outcome of the first cookie.

I hope I haven't rambled too much...

--Craig

EALR 1.4.1
> Understand the concepts of complementary, independent, and mutually
> exclusive events.

Grade Level Expectations (suggested)
> •Determine and explain when events are mutually exclusive (e.g.,
> your grade on a test is an A, B, or C). [CU, MC]
•Determine and
> explain when events are complementary (e.g., a person awake or
> asleep, you pass or fail a test, coin throw - heads or tails). [CU,
> MC]
•Identify or explain when events are complementary, mutually
> exclusive, or neither (e.g., spinning a 4 or a 5 but with the
> possibility of spinning 1, 2, 3, or 6) and explain. [CU]

EALR
> 1.4.2 Understand and apply the procedures for determining the
> probabilities of multiple trials.  

Grade Level Expectations
> (suggested)

•Calculate the probabilities of independent or
> mutually exclusive outcomes or events.
•Calculate the probability
> of an event given the probability of its complement.


Test Item
> Characteristics:
a) Items may ask students to identify mutually
> exclusive or complementary events.
b) Items may ask students to
> explain when events are complementary, mutually exclusive, or
> neither.
c) Items may ask students to list the outcomes of mutually
> exclusive or complementary events.
d) Items may ask students to
> identify or determine probabilities of experiments and situations
> including complementary or mutually exclusive events as a ratio,
> decimal, or percent.
e) Items may ask students to revise a game
> with unequal probabilities to make it a fair game.

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