Let $\pi_A = \pi_B = \frac{1}{2}$ represent the probability of selecting coin A and B respectively.
Observed data: $\{\mathbf{X_1}, \mathbf{X_2}, \mathbf{X_3}, \mathbf{X_4}, \mathbf{X_5} \}$
Unobserved: $\{Z_1, Z_2, Z_3, Z_4, Z_5 \}$
where $Z_i=1$ if $i^{th}$ sequence used coin A.
Let $h_i$ represennt number of heads in $i^{th}$ sequence.
$$\begin{align*}
P(X_i | h_i) &= P(X_i|Z_i=1,h_i)P(Z_i=1) + P(X_i|Z_i=0,h_i)P(Z_i=0) \\
P(X_i | h_i) &= \tbinom{10}{h_i} \theta_A^{h_i}(1-\theta_A)^{10-h_i} \pi_A + \tbinom{10}{h_i} \theta_B^{h_i}(1-\theta_B)^{10-h_i} \pi_B\\
P(\mathbf{X} | h_i) &= \prod_{i=1}^5 P(\mathbf{X_i}| h_i)
\end{align*}$$
$$\begin{align*}
P(X_i,Z_i | h_i) &= \big( \tbinom{10}{h_i} \theta_A^{h_i}(1-\theta_A)^{10-h_i} \pi_A \big)^{z_i}\big( \tbinom{10}{h_i} \theta_A^{h_i}(1-\theta_A)^{10-h_i} \pi_B \big)^{1-z_i} \\
\ln{P(X_i, Z_i|h_i)} &= z_i \big(\ln(\tbinom{10}{h_i}) + h_i \ln(\theta_A) + (10-h_i)
\ln(1-\theta_A) \big) +\\ & (1-z_i) \big(\ln(\tbinom{10}{h_i}) + h_i \ln(\theta_B) + (10-h_i)
\ln(1-\theta_B) \big)\\
\ln(P(X,Z|h)) &= \sum_{i=1}^n \ln(P(X_i, Z_i|h_i)\\
P(Z_i|X_i) &= \frac{P(X_i, Z_i | h_i)}{P(X_i|h_i)}
\end{align*}$$
E step
(Integration carried w.r.t $P(Z_i|X_i)$)
$$\begin{align*}
E_{Z|X}(\ln(P(X,Z))) &= E_{Z|X}[z_i] \big(\ln(\tbinom{10}{h_i}) + h_i \ln(\theta_A) + (10-h_i)
\ln(1-\theta_A) \big) +\\ & (1-E_{Z|X}[z_i]) \big(\ln(\tbinom{10}{h_i}) + h_i \ln(\theta_B) + (10-h_i)
\ln(1-\theta_B) \big)\\
E_{Z|X}[z_i] &= P(z_i=1|x_i)\\
&= \frac{\pi_A \big( \tbinom{10}{h_i} \theta_A^{h_i}(1-\theta_A)^{10-h_i} \big) }{\pi_A \big( \tbinom{10}{h_i} \theta_A^{h_i}(1-\theta_A)^{10-h_i} \big) + \pi_B \big( \tbinom{10}{h_i} \theta_B^{h_i}(1-\theta_B)^{10-h_i} \big) } \\
&= \frac{\big(\theta_A^{h_i}(1-\theta_A)^{10-h_i} \big) }{\big( \theta_A^{h_i}(1-\theta_A)^{10-h_i} \big) + \big( \theta_B^{h_i}(1-\theta_B)^{10-h_i} \big) } \cdots \ \ (1) \\
\end{align*}$$
M step
$E_{Z|X}(z_i^{(j)})$ is known at step $j$
$$
\begin{align*}
\frac{\partial E_{Z|X}(z_i^{(j)})}{\partial \theta_A} &= \sum_{i=1}^n E_{Z|X}(z_i^{(j)}) \big( \frac{x_i}{\theta_A} + \frac{1-x_i}{1-\theta_A} \big) \\
\implies \theta_A^{(j+1)} &= \frac{\sum_{i=1}^n E_{Z|X}(z_i^{(j)}) x_i}{\sum_{i=1}^n E_{Z|X}(z_i)}\cdots \ \ (2) \\
\end{align*}
$$
Walk through
1st sequence: H T T T H H T H T H
$\pi_A = \pi_B = \frac{1}{2}$
$\theta_A^{(0)} = 0.6, \theta_B^{(0)} = 0.5$
$h_i = 5$
$P(z_{i=1}^{(1)} =1 ) = \frac{\big( \theta_A^{5}(1-\theta_A)^{5} \big) }{\big( \theta_A^{5}(1-\theta_A)^{5} \big) + \big( \theta_B^{5}(1-\theta_B)^{5} \big) } = 0.4491$
$E_{Z|X}(z_i^{(1)})x_i = 0.45 *5 = 2.25 $
You can work out the other numbers similarly. Or simulate it
