| 1. |
In a production run of 10,000 logic chips, it is found that 7,325 are
defective. Based on this, what is the probability that a randomly
selected logic chip is defective? |
| 2. |
In order to determine whether a certain 6-sided die is fair, it is
rolled 200 times; the outcomes are recorded in the spreadsheet UnfairDice.xls.
Based on these outcomes, what is the probability that when the die is rolled
once, a 1 results? What is the probability that a 6 results?
What would these probabilities be if the die were fair? Based on
that, do you think the die is fair? |
| 3. |
Play the game Chuck-a-Luck a few thousand times. (Use the spreadsheet
Chuck.xls.) Based on the outcomes,
what is the probability that you lose? That you win $1? That
you win $2? That you win $3? [The game Chuck-a-Luck is played
as follows: Pick a number from 1 to 6. Roll three dice.
If none of the dice match your number, you lose $1. If exactly one
of the dice match your number, you win $1. If exactly two of the
dice match your number, you win $2. If all three dice match your
number, you win $3. At this stage, you won't be able to find the
exact probabilities; just give approximations based on the outcomes determined
by the spreadsheet.] |
| 4. |
Write the sample space for the following experiment: roll two 4-sided
dice. What is the probability that the sum of the two dice is 5? |
| 5. |
Write the sample space for the following experiment: toss four coins.
(Hint: use a tree diagram. The sample space includes: {HHHH, HHHT,
HHTH, . . .}, and has 16 elements.) What is the probability that
there are exactly two heads? |
| 6. |
Use the spreadsheet DiceIndep.xls to simulate
the rolling of a 4-sided die and a 6-sided die 1200 times. Let A
be the event that the 4-sided die gives a 3. Let B be the
event that the 6-sided die gives a 5 or 6.
| a. |
How many times did the 4-sided die give a 3? How many times did
the 6-sided die give a 5 or 6? How many times did the 4-sided die
give a 3 and the 6-sided die give a 5 or 6? |
| b. |
What is the probability of event A? What is the probability
of event B? What is the probability of event A and B? |
| c. |
In 1200 rolls, how many times (on average) would you expect the 4-sided
die to give a 3? How many times would you expect the 6-sided die
to give a 5 or 6? How many times would you expect the 4-sided die
to give a 3 and the 6-sided die to give a 5 or 6? |
|
| 7. |
Two cards are drawn from a deck, with replacement. [This means
that the first card is put back into the deck, and the deck is shuffled,
before the second card is drawn. So the drawing of the two cards
is independent.] What is the probability that both of them are aces?
What is the probability that neither of them are aces? What is the
probability of the complementary event (that at least one of them is an
ace)? |
| 8. |
Use the spreadsheet Cards.xls to simulate drawing
one card (with replacement) 1300 times.
| a. |
How many times was that card a king? How many times was that
card a heart? How many times was that card the king of hearts? |
| b. |
What is the probability that the card was a king? What is the
probability that the card was a heart? What is the probability that
the card was the king of hearts? |
| c. |
In 1300 draws, how many times (on average) would you expect to draw
a king? How many times would you expect to draw a heart? How
many times would you expect to draw the king of hearts? |
|
| 9. |
Draw a single card from a deck. What is the probability that
it is an ace or a 10? What is the probability that it is a 'face
card' (king, queen, jack) or a heart? |
| 10. |
Suppose a rocket has five separate, independent systems which all must
function correctly to prevent a launch failure. Suppose the five
systems have respective probabilities of failure of 0.03, 0.01, 0.01, 0.02
and 0.01. What is the probability that all five systems do not fail?
Would you consider the use of this launch vehicle for a critical mission? |