A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of $$ x $$ successes in the $$ n $$ independent trials of the experiment. $$ n=7, p=0.35, x

Question

A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of $$ x $$ successes in the $$ n $$ independent trials of the experiment. $$ n=7, p=0.35, x<4 $$ $$ P(X<4)=\square $$ (Round to four decimal places as needed.)

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We need to find

$$
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
$$

where, for a binomial random variable $$ X $$ with parameters $$ n = 7 $$ and $$ p = 0.35 $$,

$$
P(X = k) = \binom{7}{k} (0.35)^k (0.65)^{7-k}
$$

**Step 1. Calculate $$ P(X = 0) $$:**

$$
P(X = 0) = \binom{7}{0} (0.35)^0 (0.65)^7 = 1 \times 1 \times (0.65)^7 \approx (0.65)^7
$$

$$
(0.65)^7 \approx 0.0490
$$

**Step 2. Calculate $$ P(X = 1) $$:**

$$
P(X = 1) = \binom{7}{1} (0.35)^1 (0.65)^6 = 7 \times 0.35 \times (0.65)^6
$$

$$
(0.65)^6 \approx 0.0754, \quad 7 \times 0.35 \times 0.0754 \approx 0.1848
$$

**Step 3. Calculate $$ P(X = 2) $$:**

$$
P(X = 2) = \binom{7}{2} (0.35)^2 (0.65)^5 = 21 \times 0.1225 \times (0.65)^5
$$

$$
(0.65)^5 \approx 0.1160, \quad 21 \times 0.1225 \times 0.1160 \approx 0.2987
$$

**Step 4. Calculate $$ P(X = 3) $$:**

$$
P(X = 3) = \binom{7}{3} (0.35)^3 (0.65)^4 = 35 \times 0.042875 \times (0.65)^4
$$

$$
(0.65)^4 \approx 0.1785, \quad 35 \times 0.042875 \times 0.1785 \approx 0.2676
$$

**Step 5. Sum the probabilities:**

$$
P(X < 4) \approx 0.0490 + 0.1848 + 0.2987 + 0.2676 \approx 0.8001
$$

Thus, the probability is

$$
P(X < 4) \approx 0.8001
$$
The probability $$ P(X < 4) $$ is approximately 0.8001.

A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of \( x \) successes in the \( n \) independent trials of the experiment. \( n=7, p=0.35, x<4 \) \( P(X<4)=\square \) (Round to four decimal places as needed.)

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