Data Analysis, Statistics & Probability
Tackling No Problem with Probability
As confusing and frustrating as these items can get, they can be easily managed if you learn to employ the three-step method described below:
Step 1: OVER—Identify the number of outcomes that fit the item’s requirements.
Step 2: UNDER—Identify the total number of outcomes possible.
Step 3: Place the OVER over the UNDER. Give the item what it wants.
No Problem with Probability in Slow Motion
Now let’s look at the steps more closely, starting with a fairly common probability item.
9. If a card is pulled at random from an ordinary 52-card deck of playing cards, what is the probability of pulling a face card (Jack, Queen, or King)?
(A)
(B)
(C)
(D)
(E)
Step 1: OVER—Identify the number of outcomes that fit the item’s requirements.
Be careful: you don’t want to jump in too fast and say 3. In an ordinary deck of playing cards, each card appears one time for each of the four suits. Therefore, the total number of outcomes that fit the requirements is 12, four suits for each of the three face cards.
Step 2: UNDER—Identify the total number of outcomes possible.
If you are pulling a card at random from a 52-card deck, then there are 52 total outcomes possible.
Step 3: Place the OVER over the UNDER. Give the item what it wants.
Create a fraction with the numbers you came up with for OVER and UNDER. , answer B.
Guided Practice
Try this one on your own.
15. Count Chocula’s “trick-or-treat” bag has eight pieces of chocolate, five pieces of gum, and three jawbreakers. If two pieces are chosen at random, what is the probability that both are chocolate?
(A)
(B)
(C)
(D)
(E)
Step 1: OVER—Identify the number of outcomes that fit the item’s requirements.
Is there more than one probability here?
Step 2: Under—Identify the total number of outcomes possible.
Think carefully about all the possibilities.
Step 3: Place the OVER over the UNDER. Give the item what it wants.
Do you remember what to do to find the probability of multiple events?
Guided Practice: Explanation
This is a variation on the standard probability item. You still want to follow the basic steps, but notice how they slightly differ.
Step 1: OVER—Identify the number of outcomes that fit the item’s requirements.
Because the item asks about multiple events, we need to find the probability of each individual event first. The first time a piece of candy is picked from the bag, how many are chocolate? 8. Hold on to this number, we’ll come back to it. The second time a piece of candy is picked from the bag how many pieces of chocolate will there be (always assume the previous trials were successful)? 7. That’s it for step 1.
Step 2: UNDER—Identify the total number of outcomes possible.
This is the exact same thing we did in step 1, but now we are concerned only with the total pieces of candy. The first time a piece of candy is picked there is a total of 16 pieces, but the second time the total changes to 15. Keep these numbers handy.
Step 3: Place the OVER over the UNDER. Give the item what it wants.
Combine the numbers you came up with to create two probabilities. The probability of getting a piece of chocolate the first time is . The probability of getting a piece of chocolate the second time would be . To find the probability of these two events happening consecutively—in a row—just multiply them together:
Answer E is correct.
Independent Practice
Once you’ve completed this item, turn the page for the solution.
19. A coin is flipped three times. What is the probability that it will land on heads at LEAST once?
(A) 0
(B)
(C)
(D)
(E) 1
Independent Practice: Explanation
This item is one of the more difficult probability items you’re likely to encounter on the SAT. But you still want to stick with our trusty three-step method.
Step 1: OVER—Identify the number of outcomes that fit the item’s requirements.
All right, the stem asks for heads coming up at least once, so this means we need to include the probability of getting one head, two heads, and all three heads, as well as any of the different orders in which they can appear in three tosses. That’s a lot of thinking, and with the time constraints of the test, you’re not going to have enough time to write out all the different outcomes where you get at least one head.
Let’s try approaching this item with a little creativity. The stem asks for at least one head, but consider the opposite of this stem—What would be the probability of NO heads?—which is the same as asking for the probability of all tails. Looking at the item this way, we can identify the number of outcomes in which we get all tails. There is one chance on the first toss, one chance on the second toss, and one chance on the third toss.
Step 2: UNDER—Identify the total number of outcomes possible.
Whenever we toss a coin, there are only two possible outcomes. Because all three tosses are independent of one other—that is, the coin is picked up and tossed again—our total outcomes for each individual toss is 2.
Step 3: Place the OVER over the UNDER. Give the item what it wants.
To determine the probability of multiple events, you need to multiply the probability of each individual event together. So the probability of getting tails on all three tosses—which is the same as never getting heads—would be . Don’t be fooled by answer choice B, . The item did not ask for the probability of tossing tails three times. The item asks for the probability of tossing at least one head, which is the same as not tossing three tails. Now we subtract the probability of three tails from 1, and the result is the probability of tossing at least one head. answer D.
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