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7. Three different coins are tossed together. Find the probability of getting:

a) Exactly 2 heads
b) At least 2 heads
c) At most 2 tails

Sagot :

GinnyAnswer
Absolutely! Let's tackle this problem step by step.

### Total Possible Outcomes:
When tossing three coins, each coin has two possible outcomes: Heads (H) or Tails (T).
Thus, the total number of possible outcomes when three coins are tossed is:

[tex]\[ 2^3 = 8 \][/tex]

These possible outcomes are:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

### a) Probability of getting exactly 2 heads:

First, we need to list out the outcomes where exactly two heads appear:
- HHT
- HTH
- THH

There are 3 outcomes with exactly 2 heads.

The probability of getting exactly 2 heads is the number of favorable outcomes divided by the total number of possible outcomes:

[tex]\[ P(\text{exactly 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8} \][/tex]

So, the probability is:

[tex]\[ P(\text{exactly 2 heads}) = 0.375 \][/tex]

### b) Probability of getting at least 2 heads:

Next, we list out the outcomes where there are at least 2 heads. This includes all cases of exactly 2 heads and the case where there are 3 heads:
- HHT (2 heads)
- HTH (2 heads)
- THH (2 heads)
- HHH (3 heads)

There are 4 outcomes with at least 2 heads.

The probability of getting at least 2 heads is:

[tex]\[ P(\text{at least 2 heads}) = \frac{4}{8} \][/tex]

So, the probability is:

[tex]\[ P(\text{at least 2 heads}) = 0.5 \][/tex]

### c) Probability of getting at most 2 tails:

Finally, we count the outcomes where there are at most 2 tails. This includes all cases where there are 0, 1, or 2 tails:
- HHH (0 tails)
- HHT (1 tail)
- HTH (1 tail)
- THH (1 tail)
- HTT (2 tails)
- THT (2 tails)
- TTH (2 tails)

There are 7 outcomes with at most 2 tails.

The probability of getting at most 2 tails is:

[tex]\[ P(\text{at most 2 tails}) = \frac{7}{8} \][/tex]

So, the probability is:

[tex]\[ P(\text{at most 2 tails}) = 0.875 \][/tex]

### Summary of Results:

a) Probability of getting exactly 2 heads: [tex]\( \boxed{0.375} \)[/tex]

b) Probability of getting at least 2 heads: [tex]\( \boxed{0.5} \)[/tex]

c) Probability of getting at most 2 tails: [tex]\( \boxed{0.875} \)[/tex]
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