Math

Question Find all possible outcomes when tossing 4 fair coins, then list the ways to get at least 2 tails.

Studdy Solution

STEP 1

Assumptions
1. A fair coin has two possible outcomes when tossed: heads (H) or tails (T).
2. We are considering the tossing of four fair coins.
3. We will use a tree diagram to represent all possible outcomes.
4. We are interested in listing all outcomes that have at least two tails.

STEP 2

To start the tree diagram, we consider the first coin toss. There are two possible outcomes: heads (H) or tails (T).

STEP 3

For each outcome of the first coin toss, the second coin toss also has two possible outcomes. This results in a total of 2×2=42 \times 2 = 4 outcomes after two coin tosses.

STEP 4

We continue this pattern for the third and fourth coin tosses. Each outcome of the second coin toss leads to two possible outcomes for the third coin toss, resulting in 4×2=84 \times 2 = 8 outcomes after three coin tosses.

STEP 5

Finally, each outcome of the third coin toss leads to two possible outcomes for the fourth coin toss, resulting in 8×2=168 \times 2 = 16 outcomes in total for all four coin tosses.

STEP 6

Now we will list all 16 possible outcomes in the form of a tree diagram. Each level of the tree represents a coin toss, and each branch represents an outcome of that toss.

STEP 7

The first level of the tree diagram has two branches: H and T.

STEP 8

The second level of the tree diagram has four branches: HH, HT, TH, TT.

STEP 9

The third level of the tree diagram has eight branches: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

STEP 10

The fourth and final level of the tree diagram has sixteen branches, which are the complete outcomes for all four coin tosses: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

STEP 11

Now we will list all the outcomes that have at least two tails. We will look for outcomes that have two or more T's in them.

STEP 12

List the outcomes with exactly two tails: HHTT, HTHT, HTTH, THHT, THTH, TTHH.

STEP 13

List the outcomes with exactly three tails: HTTT, THTT, TTHT, TTTH.

STEP 14

List the outcome with four tails: TTTT.

STEP 15

Combine the lists from steps 12, 13, and 14 to have the complete list of outcomes with at least two tails.
The ways of getting at least two tails when four fair coins are tossed are: HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT.

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