Answer:
Let X be the event of feeling secure after saving $50,000,
Given,
The probability of feeling secure after saving $50,000, p = 40 % = 0.4,
So, the probability of not feeling secure after saving $50,000, q = 1 - p = 0.6,
Since, the binomial distribution formula,
![P(x=r)=^nC_r p^r q^{n-r}](https://tex.z-dn.net/?f=P%28x%3Dr%29%3D%5EnC_r%20p%5Er%20q%5E%7Bn-r%7D)
Where, ![^nC_r=\frac{n!}{r!(n-r)!}](https://tex.z-dn.net/?f=%5EnC_r%3D%5Cfrac%7Bn%21%7D%7Br%21%28n-r%29%21%7D)
If 8 households choose randomly,
That is, n = 8
(a) the probability of the number that say they would feel secure is exactly 5
![P(X=5)=^8C_5 (0.4)^5 (0.6)^{8-5}](https://tex.z-dn.net/?f=P%28X%3D5%29%3D%5E8C_5%20%280.4%29%5E5%20%280.6%29%5E%7B8-5%7D)
![=56(0.4)^5 (0.6)^3](https://tex.z-dn.net/?f=%3D56%280.4%29%5E5%20%280.6%29%5E3)
![=0.12386304](https://tex.z-dn.net/?f=%3D0.12386304)
(b) the probability of the number that say they would feel secure is more than five
![P(X>5) = P(X=6)+ P(X=7) + P(X=8)](https://tex.z-dn.net/?f=P%28X%3E5%29%20%3D%20P%28X%3D6%29%2B%20P%28X%3D7%29%20%2B%20P%28X%3D8%29)
![=^8C_6 (0.4)^6 (0.6)^{8-6}+^8C_7 (0.4)^7 (0.6)^{8-7}+^8C_8 (0.4)^8 (0.6)^{8-8}](https://tex.z-dn.net/?f=%3D%5E8C_6%20%280.4%29%5E6%20%280.6%29%5E%7B8-6%7D%2B%5E8C_7%20%280.4%29%5E7%20%280.6%29%5E%7B8-7%7D%2B%5E8C_8%20%280.4%29%5E8%20%280.6%29%5E%7B8-8%7D)
![=28(0.4)^6 (0.6)^2 +8(0.4)^7(0.6)+(0.4)^8](https://tex.z-dn.net/?f=%3D28%280.4%29%5E6%20%280.6%29%5E2%20%2B8%280.4%29%5E7%280.6%29%2B%280.4%29%5E8)
![=0.04980736](https://tex.z-dn.net/?f=%3D0.04980736)
(c) the probability of the number that say they would feel secure is at most five
![P(X\leq 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)](https://tex.z-dn.net/?f=P%28X%5Cleq%205%29%20%3D%20P%28X%3D0%29%20%2B%20P%28X%3D1%29%20%2B%20P%28X%3D2%29%20%2B%20P%28X%3D3%29%20%2B%20P%28X%3D4%29%20%2B%20P%28X%3D5%29)
![=^8C_0 (0.4)^0(0.6)^{8-0}+^8C_1(0.4)^1(0.6)^{8-1}+^8C_2 (0.4)^2 (0.6)^{8-2}+8C_3 (0.4)^3 (0.6)^{8-3}+8C_4 (0.4)^4 (0.6)^{8-4}+8C_5(0.4)^5 (0.6)^{8-5}](https://tex.z-dn.net/?f=%3D%5E8C_0%20%280.4%29%5E0%280.6%29%5E%7B8-0%7D%2B%5E8C_1%280.4%29%5E1%280.6%29%5E%7B8-1%7D%2B%5E8C_2%20%280.4%29%5E2%20%280.6%29%5E%7B8-2%7D%2B8C_3%20%280.4%29%5E3%20%280.6%29%5E%7B8-3%7D%2B8C_4%20%280.4%29%5E4%20%280.6%29%5E%7B8-4%7D%2B8C_5%280.4%29%5E5%20%280.6%29%5E%7B8-5%7D)
![=0.6^8+8(0.4)(0.6)^7+28(0.4)^2(0.6)^6+56(0.4)^3(0.6)^5+70(0.4)^4(0.6)^4+56(0.4)^5(0.6)^3](https://tex.z-dn.net/?f=%3D0.6%5E8%2B8%280.4%29%280.6%29%5E7%2B28%280.4%29%5E2%280.6%29%5E6%2B56%280.4%29%5E3%280.6%29%5E5%2B70%280.4%29%5E4%280.6%29%5E4%2B56%280.4%29%5E5%280.6%29%5E3)
![=0.95019264](https://tex.z-dn.net/?f=%3D0.95019264)