3. Your friend makes a free throw $$ 50 \% $$ of the time. Your friend attempts four free throws, You design and perform a simulation that involves tossing four coins. Heads $$ (H) $$ represents making the free throw, and tails $$ (T) $$ represents not making the free throw.

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1. Since each free throw is independent with a $$ 50\% $$ chance of success, the situation can be modeled by four independent coin tosses where heads $$ (H) $$ indicates a made free throw and tails $$ (T) $$ indicates a missed free throw.

2. The total number of outcomes when tossing four coins is  
   $$
   2^4 = 16.
   $$

3. To find the probability of making exactly $$ k $$ free throws (i.e. obtaining $$ k $$ heads), we use the binomial formula:  
   $$
   P(X=k)=\binom{4}{k}(0.5)^k(0.5)^{4-k} = \binom{4}{k}\frac{1}{16}.
   $$

4. Calculating the probabilities for each possible value of $$ k $$:

- For $$ k=0 $$ (no free throws made):
     $$
     P(X=0)=\binom{4}{0}\frac{1}{16}=1\cdot\frac{1}{16}=\frac{1}{16}.
     $$
     
   - For $$ k=1 $$ (one free throw made):
     $$
     P(X=1)=\binom{4}{1}\frac{1}{16}=4\cdot\frac{1}{16}=\frac{4}{16}=\frac{1}{4}.
     $$
     
   - For $$ k=2 $$ (two free throws made):
     $$
     P(X=2)=\binom{4}{2}\frac{1}{16}=6\cdot\frac{1}{16}=\frac{6}{16}=\frac{3}{8}.
     $$
     
   - For $$ k=3 $$ (three free throws made):
     $$
     P(X=3)=\binom{4}{3}\frac{1}{16}=4\cdot\frac{1}{16}=\frac{4}{16}=\frac{1}{4}.
     $$
     
   - For $$ k=4 $$ (all four free throws made):
     $$
     P(X=4)=\binom{4}{4}\frac{1}{16}=1\cdot\frac{1}{16}=\frac{1}{16}.
     $$

5. The simulation that involves tossing four coins will generate outcomes equivalent to these probabilities. If you repeat the simulation many times, the relative frequencies of obtaining 0, 1, 2, 3, or 4 heads should approximate the theoretical probabilities calculated above.
When your friend attempts four free throws with a 50% chance of making each, the probability of making exactly $$ k $$ free throws is calculated using the binomial formula. Here are the probabilities for each possible number of successful free throws: - 0 made: 6.25% - 1 made: 25% - 2 made: 37.5% - 3 made: 25% - 4 made: 6.25% These probabilities are based on tossing four coins, where heads represent a made free throw and tails represent a missed one.

3. Your friend makes a free throw \( 50 \% \) of the time. Your friend attempts four free throws, You design and perform a simulation that involves tossing four coins. Heads \( (H) \) represents making the free throw, and tails \( (T) \) represents not making the free throw.

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